The filming on Harry Potter & The Prisoner of Azkaban will start sometime between January and March 2003.
Cuaron is no stranger to directing children's movies - he directed The Little Princess in 1995.
Chris Columbus, who directed the first two Potter movies said he's 'thrilled' about the choice. Columbus will also help produce the film.
Cuaron said: "I am so pleased to be entrusted with presenting the continuing saga of Harry Potter."
Azkaban should be in cinemas in exactly two years time in summer 2004.