Depp will play the Big Bad Wolf and Streep will play The Witch.
Rob Marshall (Pirates of the Caribbean: On Stranger Tides, Chicago) is to direct.
The announcement was made last weekend at Disney's biggest fan event, their D23 Expo, at which many star actors made appearances. Among them was British talent Tom Hiddleston, who performed Jungle Book song The Bare Necessities (see video here).
Sondheim's award-winning 1987 musical, which has seen numerous revivals on both sides of the Atlantic, weaves together the characters from several Brothers Grimm fairy tales, including Jack and the Beanstalk, Little Red Riding Hood, Rapunzel and Cinderella.
Though this is by no means Depp's first foray into the musical on film, following his portrayal of the titular character in Tim Burton's Sweeney Todd in 2006 (also by Sondheim), he still has yet to make the final leap and take to the stage.
According to the actor Steven Berkoff, who appeared with Depp in the 2010 film The Tourist, this is because the Pirates of the Caribbean star is "afraid of hitting the wall on stage and forgetting his lines".
And if he is considering retiring in the next three years, as recently reported, there may not be much time left.
The film of Into The Woods is due for release in 2014.
- Alex Gordon