What follows is a very literal translation from the Hebrew translation because Rebbe Nachman produced his precious stories with the intention that every word is rife with meaning and a less literal translation loses some of those meanings.
(Even though any translation at all, by its very nature, loses some of the original Hebrew or Yiddish or Aramaic meaning, although when translating between these languages, very little is lost due to their similarities in linguistics and spirit.)
You can find smoother translations online and in books.
The words in curved parentheses () are also found in the original transcription of the story. The words in square parentheses  are my own additions: translations or the Hebrew original for clarification, or the Malbim's definition of that original Hebrew word.
(For example, the Malbim defines bakasha/mevakesh as "requesting one's heart's desire.")
Then follows a few commentaries on this story by Breslov Sages from past generations.
Then a list of gemstones and their segulot [virtues/qualities] follows, in case you were wondering what were some of these segulot that the king and queen imbibed with the wine.
It's a long post, and you may only be interested in parts of it, so I tried to label things clearly so that you can skip around and read only what's useful to you.
The Prince Made Entirely of Precious Gems
And he decreed upon the Jews that they shall pray from him that he shall have children. And the Jews were requesting [their heart’s desire] and seeking a tzaddik in order that he shall pray and [actively] facilitate that he shall have children, and they requested [their heart’s desire] and found a hidden tzaddik and they said to him that he should pray that the king shall have children; and he answered that he doesn’t know anything.
And they made this known to the king, and the king sent his imperial decree after him and they brought him to the king.
And the king began to speak with him with goodness: “Do you not know that the Jews they are in my hand to do with them according to my desire, therefore I request from you in goodness that you will pray that there will be children for me.”
And he promised him that there shall be for him in the same year a child [vlad], and he went to his place.
And the queen bore a daughter, and this same daughter of the queen was of very beautiful form [yafat to'ar me'od - specifically the shape of the face] and when she was 4-year-old, she was skilled in all the wisdoms and to play singing instruments, and she knew all the languages, and kings from all the countries would journey to see her, and a great [perpetual state of spiritual] joy was upon the king.
Later, the king yearned greatly that he shall have a son in order that his kingdom shall not be turned over to a foreign man, and he decreed again upon the Jews that they shall pray that he shall have a son. And they were requesting [their heart’s desire] and seeking after for the tzaddik mentioned above, and they didn’t find him, because he already passed away.
And they requested [their heart’s desire] more, and found another hidden tzaddik and they said to him that he shall give to the king a son, and he said that he doesn’t know anything.
And they made known to the king, and the king said to him also as mentioned above: “Do you not know the Jews are in my hand, etc., as mentioned above.
The wise man (meaning, the tzaddik mentioned above) said to him: “Are you able to do what I shall command?”
The king said, “Yes.”
The wise man said to him: “I need that you shall bring all the types of good gemstones because every good gemstone, it has another segulah [virtue/quality]” -- because there is by the kings a book in which is written all types of the good gemstones.
The king said: “I will take out half my kingdom in order that a son will be for me.”
And he went and brought for him all types of the good gemstones.
And the wise man took them and ground them, and took a cup of wine and passed them to within it, and gave half the cup to the king to drink and half of it to the queen; and he said to them that there shall be for them a son that all of him shall be from good gemstones, and all the segulot of all the good gemstones, and he went to his place.
And she bore a son, and a great [perpetual state of spiritual] joy was made upon the king. And the son born was not from good gemstones. When the son was four years old, he was of very beautiful form [yafeh toar – specifically the shape of the face] and a great wise one in all the wisdoms, and he was knowing all the languages, and they would travel to the king to see him.
And the daughter of the queen saw that she isn’t so important, and became envious [believed he will diminish or take some benefit or purpose that she believes she deserves] of him; only this was her consolation: in that which the same tzaddik said that all of him will be from good gemstones—good that he is not from good gemstones.
One time, the son of the king was carving up [m’chatech] trees and he was wounded in his finger, and the daughter of the queen wanted to swathe/wrap/bandage his finger, and she saw there a good gemstone, and she became very envious [believed he will diminish or take some benefit or purpose that she believes she deserves] of him. And she made herself sick, and some doctors came, and they were not able to make for her a healing.
And they called for the sorcerers, and there was a sorcerer, and she revealed to him the truth, that she made herself sick as mentioned above. And she asked him if he would be able to perform sorcery to a person that he shall be leprous [metzora].
He said: “Yes.”
She said to him, “Perhaps a sorcerer will request [his heart’s desire] that the spell will be cancelled, and he will be healed.”
The sorceror said: “If they will cast the spell to the water—they will not be able to cancel it anymore."
And she did so, and she cast the spell to the water, and the son of the king was made very leprous [metzora me'od], (and he had) on his nose [chotmo] leprosy [tzaraat] and upon his face and upon the rest of his body, and the king busied with doctors and with sorcerers—and they weren’t useful.
And he decreed on the Jews that they shall pray, and they requested [their heart’s desire] the tzaddik mentioned above and they brought him to the king and the tzaddik mentioned above was praying always to Hashem Yitbarach in that that he was promising that the son of the king will be all of him from good gemstones—and it was not so.
And he was contending to Hashem Yitbarach: “Did I do this for my kavod [inherent beauty—i.e. to make me look good]? I did not do it but for Your Kavod [inherent beauty]; and now it did not come into existence/to fruition like I said.”
And the tzaddik came to the king, and he was praying—and it wasn’t effective, and they made known to him that it was a spell; and the tzaddik mentioned above was higher above all the sorcerers.
And the tzaddik came to the king and made known to the king that it is a spell; and that they cast the spell to the water, and there is no remedy for the son of the king—except that they shall cast the sorcerer who made the spell to the water.
The king said: “I give to you all the sorcerers to cast them to the water in order that my son will be healed.”
And the daughter of the queen became afraid [fear of a known thing], and she ran to the water to take out the spell from the water, because she had known where the spell lay; and she fell to the water, and a great commotion was made because the daughter of the king fell to the water.
And the same tzaddik mentioned above came and said to them that the son of the king shall be healed and he became healed. And the leprosy became dry and fell, and all the skin became peeled from him, and he was made of good gemstones all of him, and he had all the segulot of all the good gemstones (meaning, that after the skin peeled, then it became revealed that the son of the king, he consists entirely of good gemstones, as the tzaddik mentioned above said).
It is evident to every one who probes it that this story is awesome and wondrous – and by the way hints: "Hashem is with me, I won't fear; what can Man do to me?...Many plans are in the heart of Man, but only Hashem's counsel will come to fruition" because davka via what the princess did to him with occult powers he was made leprous and through that, he was healed and afterwards made completely of precious gems, etc.
And it is also wonderful encouragement that sometimes a descent is the reason for an ascent. Aside from the many secrets woven in this story, and that contained within is the whole Holy Name of 42 letters, and I heard this story is also about why the known thing fell sometimes in the water and almost drowned, etc.
Zimrat HaAretz [Song of the Land] by the Tcheriner Rav
There are those who say this story's meaning was heard from Rabbeinu [Rebbe Nachman] that contained within is the secret 42-letter name, etc., as is explained in another place [Zohar Chadash, the end of Parshat Ki Tavo]....Also what's explained here that the prince became leprous and afterwards, he was healed and the scabs dried up and all the skin peeled off and then it was revealed and seen that he was comprised entirely of precious gems and he had the special properties of each gem.
We find in this an inyan of inheriting Eretz Yisrael. Like the Sages said regarding the verse: "When you will come to the Land of Canaan and I will give you a blemish of tzaraat [leprosy] in the house of the Land of your inheritance," like Rashi explains there that through those means, they found in Israel hidden treasures of gold that the Emorites hid in the walls of their homes.
And in Midrash Rabbah, our Sages explained this verse to refer to the destruction of the Beit Hamikdash. And it concludes there that they took other stones as it says, "Behold I am established in Tzion stone by stone, I scrutinize the corner of that which is precious, etc." And there lay the foundation stone that comprised all the 12 gems of Yaakov Avinu, which symbolize the 12 Tribes who contained within them the special properties of each precious gem. And that means the 12 gems of the Choshen as our Sages expounded in Bresheit Rabbah 17, regarding the bedolach, the shoham regarding Eretz Yisrael.
Sichot HaRan [Conversations of Rebbe Nachman] by Rebbe Nosson Sternhartz (1780-1844)
I heard from one of our men who said that before Rabbeinu told this story, he said, "I know a story which contains the entire 42-letter name." And then he told this story.
Nonetheless, we don't know if this is the story containing the 42-letter name. Furthermore, I heard from his holy mouth a few years ago that he said that the Besht knew a story which contained the 42-letter name and spoke with me then of the 42-letter name. And he asked me to find an explanation in non-Hebrew regarding 2 letters – Vav and Tzadi – which are found in the above-mentioned name, and I couldn't find them. Since it was evident that he already knew the secret of the whole name, only those 2 letters, he still couldn't incorporate them into the matter when he wanted to clothe it in that name.
Leviat Hachen (Rav Levi Yitzchak Bender's commentary)
Once upon a time, there was a king who had no children...and decreed that the Jews should pray for him – The nations know well that the Jews have the power to pray and bring healing and salvation through the power of their prayer. But because of the bad traits found in each of the 70 nations, they always want to uproot Yisrael from its roots. Yet when they undergo trials and tribulations, the come to the Jews so that the Jews will pray for them because in a time of trouble, the truth reveals itself in all its glory.
A Tzaddik is One Whose Prayer is Set in His Mouth
And the Jews were seeking and searching a Tzaddik who would pray and set things in motion for the king to have children – In Likutei Moharan, Rabbeinu writes that Hashem loves and strongly desires the prayers of tzaddikim and habituates continual (set, fixed) prayer in their mouth. Therefore, they sought out a tzaddik whose prayer is routine (continuous) and set in his mouth so that when he prays, he is answered, and that's how it was.
When You Search, You Find
And they sought and found a hidden Tzaddik – One who seeks will find. As it says: "You strove and you found – believe..."
Through Self-Nullification, Prayer is Accepted
And he answered that he doesn't know at all – That is the way of a Tzaddik. He upholds perfect humility, meaning, "What" – as if he doesn't know anything and he completely nullifies himself to Hashem, then he can set things in motion through his prayer.
The Tzaddik's Entire Being is Via Dignified Reclusion – and When He becomes Renowned, He Disappears
And they sought and found a hidden Tzaddik...and the king began to speak with him about good things...and he promised him that he would have a child that same year...and the queen birthed a daughter...Afterwards, the king yearned for a son...and they were seeking and searching for the same tzaddik, but they didn't find him because he had already passed away. And they found another hidden tzaddik...and he told them that they'll have a son who will be made entirely of precious gems – Breslevers used to say here that there are tzaddikim that their entire reason for existence is only in hiding and dignified reclusion, and immediately – when they are revealed in This World – they are forced to die and disappear from This World. And therefore, when his promise was fulfilled and a daughter was born, he passed away. The second Tzaddik promised the king a son made of precious gems, but when the son was born, the precious gems couldn't be seen, therefore, he could continue living because his promise wasn't completely fulfilled.
The Extent of Envy and Resentment
The Daughter of the Queen saw that she was no longer so esteemed and she became envious of him – Take note of the extent of the Yetzer's power over jealousy and resentment. Solely because of her envy, she went to dabble in occultism in order to harm her brother [but the Guardian of Israel neither sleeps nor slumbers]. And the strongest envy is by the greatest ones, like the daughter of the queen, who as the only princess was highly esteemed and educated, and kings and emperors came to see her, and now – because of envy – she fell and did what she did until in the end, she fell in the water, may G-d have mercy.
The Precious Gem is Revealed After the Foul Blood Drains Out
And he was struck on his finger, and the queen's daughter wanted to bandage his finger when she saw there a precious gem – The Breslevers used to point out here that usually, when a finger is struck, it bleeds; that is to say that beforehand, there was foul blood that covered the precious gems. And therefore, later, when his scabs dried off, it was revealed that his entire being was made of precious gems. Rebbe Shmuel Eizik was one of the Gedolim who served Hashem of the talmidim of Rabbeinu. Once, Rabbeinu grabbed close to his heart and said, "For a quarter-cup of blood that lies in the chamber of the heart, you could lose two world at the same time – increase upon it groans and sighs and tame it. And you will merit the status: "And my heart is a hollow chamber within me."
A Little Bit of Sacrifice and One Already Sees the Precious Gem
And he was struck on his finger...when she saw there a precious gem – The draining of the blood via the blow to the finger indicates self-sacrifice. And even through a little bit of self-sacrifice, one already sees the precious gem. For every Jew contains within a precious gem, only that gem is covered and invisible. And through a little bit of self-sacrifice – sleep a little less, eat with a bit less taavah and the rest of the matters of kedushah – that precious gem is revealed.
Gemstones and their Segulot
He wrote it out according to the gemstones representing each Tribe on the Choshen, so that's how it appears here too:
Reuven – Ruby (Arabic: “Rubin” and is from the same family as the blashu stone, but is of better quality in color) – Every woman who holds this stone will not ever miscarry; and it’s good for a woman with difficult labors; and if it’s finely ground and added to food or drink, it enhances fertility like dudaim.
Shimon – [Emerald-smaragd?] Markadi, common to East Africa – a higher quality gem in radiance and sparkle and color than the “prashma,” although both are the same and the markadi is cut from the prashma/beryl) – It cools one’s taavah.
Levi – Chalcedony (“Carbuncle,” “Firestone”-- a kind of opal) – Grants wisdom to the simple and enlightens one’s eyes. Ground finely and added to food, drink, or drug, it grants wisdom and opens one’s heart.
Yehudah – [Higher-quality emerald?] Zamura – When one carries it, all one’s enemies flee.
Yissachar – Light Sapphire – It’s good for the light of one’s eyes when one passes it over one’s eyes; it relieves every pain and swelling in every place.
Zevulun – Pearl – It brings sleep upon a person
Dan – Topaz, Ashtapsis
Naftali – Turquoise – Horseriders make sure to keep one when they journey. It pulls a man along in his vehicle and makes his “riding all his days” successful.
Gad – Crystal, a common stone owned by many and known by many; it strengthens one’s heart when one goes to war.
"When worn on one’s finger, it enables one to see dreams." [Ibn Ezra, Shemot 28:9]
Asher – [Chrysolite\Peridot?] (kriulika) – It’s the color of oil although some say it’s light blue; it assists with digestion and when you ground it finely and mix it with food, it thickens and enriches the food.
Yosef – Onyx, Unikli – It grants a man chen [favor] in the eyes of all, and whoever carries it in a royal house will be greatly benefited and succeed and all his words will be obeyed.
Binyamin – Jasper, Yashfiz, red-black-green – This stone has the power to stop bleeding\hemorrhage.