Dean Corso (Johnny Depp), a New York City rare book dealer, makes his living conning people into selling him valuable antique books for a low price, and then re-selling them to private collectors. Corso meets with wealthy book collector Boris Balkan (Frank Langella), who has recently acquired a copy of the (fictional) book The Nine Gates of the Kingdom of Shadows by 17th-century author Aristide Torchia, one of only three extant copies. The book is an adaptation of one written by the Devil himself and purportedly contains the means to summon the Devil and acquire invincibility and immortality. Balkan believes two of the copies are forgeries. He hires Corso to check all three, and acquire the legitimate one by any means necessary.
Balkan's copy was acquired from Andrew Telfer (Willy Holt), who killed himself soon after. Telfer's widow Liana (Lena Olin) seduces Corso, in a failed attempt to get the book back. Meanwhile, Corso leaves the book for safekeeping with bookseller Bernie Rothstein (James Russo), who is then murdered; his corpse is found posed like an engraving in The Nine Gates.
Corso travels to Toledo, Spain. The Ceniza brothers, book restorers, show him that three of the engravings are signed "LCF". Corso deduces that Lucifer himself designed and cut them. Corso travels to Sintra, Portugal, to compare Victor Fargas' (Jack Taylor) copy of the book to Balkan's. To Corso's surprise, he discovers that the signature "LCF" is found in three different engravings, which vary in small but significant details from the images in the Balkan copy. The next morning, a mysterious young woman (identified only as "the Girl") (Emmanuelle Seigner) who appears to have been shadowing Corso since Balkan hired him, awakens Corso and leads him to Fargas' house. He finds the old man murdered and the "LCF"-signed engravings ripped out of that copy.
In Paris, Corso visits the Baroness Kessler (Barbara Jefford), who owns the third copy. At first the Baroness refuses to cooperate, but Corso intrigues her with evidence that the engravings differ among the three copies. He explains his idea: each copy contains three different "LCF"-signed engravings, therefore all three copies are required for the ritual. Corso finds "LCF" on three different engravings in the Baroness's book confirming his theory.
Kessler is killed, and the Girl rescues Corso from Liana's bodyguard. When Liana steals Balkan's copy from Corso's hotel room, he follows her, and witnesses her using the book in a Satanist ceremony. Balkan suddenly interrupts the ceremony, kills Liana, and leaves with the engraved pages and his own intact copy.
Corso pursues Balkan to a remote castle, depicted in one of the engravings, and finds Balkan preparing the final ritual. After a struggle, Balkan traps Corso in a hole in the floor. Balkan performs his summoning ritual: he arranges the engravings on a makeshift altar, and recites a series of phrases related to each of the nine engravings. Balkan then douses the floor and himself with gasoline and sets it alight, believing himself to be immune to suffering. Balkan's invocation fails, and he screams in pain as the flames engulf him. Corso frees himself, kills Balkan, takes the engravings, and escapes.
Outside, the Girl appears and has sex with him by the light of the burning castle. She tells him that Balkan failed because the ninth engraving he had used was a forgery. On her suggestion, Corso returns to the Ceniza brothers' now vacant shop. By chance he finds the authentic ninth engraving. On it, there is a likeness of the Girl. With the last engraving in hand, Corso returns to the castle. He has completed the ritual requirements, and he crosses through the Ninth Gate into the light.

Question: What does Boris Balkan hire Dean Corso to do?
Answer: Research three copies of the book The Nine Gates of the Kingdom of Shadows to find the original copy.
Alternative answer: to locate all three copies of the book, two of which he believes to be forgeries, and to acquire the legitimate book