The 30-year-old star - who plays the titular Time Lord in the sci-fi series - has been cast as murderous Wall Street investment banker Patrick Bateman in the production, based on Bret Easton Ellis's 1991 novel, which will run at the Almeida Theatre from December 3 to January 25, 2014.
The musical is to be directed by Rupert Goold - the theatre's new artistic director - who developed the production with his former company Headlong.
Tony Award-winning composer Duncan Sheik has written the songs for the show, while the book was adapted into a script by Roberto Aguirre-Sacasa - whose previous work includes Spider-Man: Turn Off the Dark.
Ellis's novel was previously made into a 2001 film starring Christian Bale as sadistic serial killer Bateman with Reese Witherspoon, Willem Dafoe, Jared Leto and Chloe Sevigny in supporting roles.
Smith recently filmed his final scenes as the Doctor for the upcoming Doctor Who Christmas special, which will be his last episode.
He announced in he was leaving the BBC show in June and he will be replaced in the TARDIS by Peter Capaldi.
