On a recent Saturday afternoon, visitors wandered through Sotheby’s, on the Upper East Side, to inspect articles featured in this season’s auctions. On the ground floor, they could inquire about a bottle of Château Mouton Rothschild 1945 from the wine cellar of the businessman William Koch. In the exhibition halls upstairs, a Mayan classical-period stucco head was on display, as were a collection of Signacs and two diminutive Francis Bacon self-portraits. And, on the sixth floor, in a small room outside the jewelry section, surrounded by images of the cosmos and encased in glass, a truly unusual object was rotating atop a velvet-covered pedestal: the Lesedi La Rona, the biggest diamond in the world. At eleven hundred and nine carats, the stone was about the size of a fist. Its skin was opaque in places, but certain angles offered brief, clear glimpses into the depths of the crystal: a tightly woven, perfectly symmetrical carbon lattice.

Lucara Diamond, a Vancouver-based mining company, was struggling to balance the books in 2013 when it began finding large stones at its only operating mine, in Botswana. Then, last November, the mine’s X-ray machine found a clod of kimberlite that an underground volcanic eruption had shot to the Earth’s surface, like a hundred-and-fifty-pound soccer ball, some ninety million years ago. Lucara’s workers picked away the debris, washed off the dust, and found the biggest gem-quality diamond recovered in a hundred years.

Traditionally, rough diamonds are sold via private bids. Prospective buyers come to inspect the stone, try to imagine what they can cut it into and what that could sell for, and then make offers. But, given the singular nature of this find, Lucara decided on a different approach, hiring Sotheby’s to put the diamond up for public auction and organizing a competition in Botswana to name it. The winner, Lesedi La Rona, was chosen from among eleven thousand entries; it means “our light” in Setswana. Soon the stone was on a world tour, for viewings in Hong Kong, Dubai, New York, Geneva, and, finally, London, where it will go under the gavel on June 29th.

Polished gems go up for auction all the time, but rough diamonds are almost never sold in this way. David Bennett, the chairman of Sotheby’s jewelry division, told me that he put a piece of rough pink diamond up fifteen years ago, but it didn’t sell. The prevailing assumption is that no one outside the industry would know what to do with a huge rough diamond—or how to properly value it. Because it’s impossible to put a stone back together again once it has been cut, and because each one is unique, the value of a rough diamond is unpredictable even when it’s small. The Lesedi La Rona’s enormous size makes its worth even harder to assess.

The going rate for an exceptionally large rough diamond at the time of the discovery was about sixty thousand dollars per carat, which would have put the stone's value above sixty million dollars. But, in a private sale early this May, Lucara sold an eight-hundred-and-thirteen-carat stone of similar quality, which it found in the same mine the day after it recovered the Lesedi La Rona. The smaller diamond went for sixty-three million dollars, or more than seventy-seven thousand dollars per carat. With that sale as a benchmark, Sotheby’s now estimates that the Lesedi La Rona is worth at least seventy million dollars, but this could prove low, given the appeal (and challenge) of owning the world’s largest diamond, combined with the decision to sell it at auction, which has expanded the pool of potential buyers to include fine-art collectors as well as diamantaires.

In an effort to draw interest from outside the diamond business, Sotheby’s brought in two experts to analyze the stone and write reports, to be shown exclusively to prospective buyers, describing how it might be turned into polished gems. One of the experts was Ronnie VanderLinden, a veteran of Manhattan’s diamond district, on Forty-seventh Street. He told me that he spent hours turning the stone from one side to the next, photographing, louping, microscoping. In recent decades, it has become standard practice to run stones through M.R.I. machines, to map their dimensions along with any cracks, smaller feathers, or inclusions (specks of foreign substances, usually graphite, within the stone), but the Lesedi La Rona didn’t fit into existing scanners. “It’s like the old days,” VanderLinden said. “We’re relying on our two eyes.”

Assessing a rough diamond requires a geometrical imagination, as well as experience with the behavior of the crystals, which have a grain that can affect the way a diamond reacts to the cutting process. With large, complex stones, cutters often work slowly. Polish the stone a little, put it away, wake up in the middle of the night with an idea. Cutters describe listening to a stone’s “song”—to the way the grain interacts with a polishing scaife. Lasers now allow cutters to bend the grain, expanding the number of angles they can take. Shaping the Lesedi La Rona would be a long chess game, and you couldn’t be certain what you had until you had it.

The only known gem-quality stone larger than the Lesedi La Rona was discovered, in 1905, outside Pretoria, South Africa, and sold to the British-controlled colonial government for a hundred and fifty thousand pounds (more than sixteen million pounds, or twenty-five million dollars, today). Named the Cullinan, after the chairman of the company that found it, the rough stone was nearly three times the size of the Lesedi. It was of the highest clarity over all, but there were visible inclusions in its center. To work around them, the diamond was divided into nine main stones, and more than a hundred smaller pieces, before being polished. The largest stone became the five-hundred-and-thirty-carat Great Star of Africa, which was placed in the orb of the royal sceptre in the Tower of London. William Lamb, Lucara’s C.E.O., told me that, because of its shape and clarity, the Lesedi La Rona could, if skillfully cut, yield an even larger stone—the largest polished diamond in history.

The story behind a diamond—its pedigree, provenance, and history—is integral to its worth, and the tale of one that could outshine the Queen of England’s will have particular appeal to some prospective bidders. Chiefly responsible for telling it at Sotheby’s is Bennett. As the chairman of the jewelry division, he has handled the sale of stones that were owned by the Duchess of Windsor, Ava Gardner, Maria Callas, and the family of the Aga Khan, and he has repeatedly broken price records. Last year he sold the Blue Moon, a twelve-carat blue diamond, to the Hong Kong billionaire Joseph Lau for forty-eight million dollars, the highest anyone had paid for a single polished stone.

Bennett draws on broad expertise to accentuate the mystique of exceptional stones. On top of the history of jewelry, he has studied alchemy, the Hermetic sciences, and medieval astrology. When I asked where diamonds fit into such systems, he said that they were always central, associated with the sun. He recalled the coronation of Marie de Medici, as depicted by the artist Peter Paul Rubens. The diamond-encrusted crown, he surmised, would have been placed on her head at noon, when the gems would most dazzle onlookers. He described the Lesedi La Rona, a single piece of rough some three billion years old, as “almost ungraspable.”

The final price the diamond draws at auction will be determined, in part, by what the buyer wants to do with it. Although Sotheby’s had been considering its potential value as a polished gem, William Lamb told me that he would like to see the buyer keep the rough stone as it is—as a collector’s item unto itself. He described possessing it as “owning a piece of creation.” For an ancient sculpture, a piece of Precambrian art, what is ninety million dollars? If Cy Twombly’s blackboard painting, an arrangement of “oil based house paint and wax crayon on canvas,” can draw nearly thirty-seven million dollars at auction, the Lesedi La Rona might be a bargain. Although, unlike Twombly, the diamond mines are still at work. You never know when a bigger stone might turn up.