The duo, who teamed for the successful films Edward Scissorhands, Sleepy Hollow and Ed Wood, among others, are now working on a cinema adaptation of the Stephen Sondheim musical.
Depp will play Todd, the legendary London barber who murdered his clients by slitting their throats.
The musical opened on Broadway in 1979, and has earned eight Tony Awards.
Filming for Sweeney Todd begins early next year.