Spacey will play the villainous English king in a production of Shakespeare's play directed by Mendes, which is set for an international tour in 2012.
The play will form the third and final season of the Bridge Project, which brings together British and American actors on classic plays directed by Mendes.
"Richard III" will open at New York's Brooklyn Academy of Music in February 2012, then tour internationally before opening that May at London's Old Vic Theatre, where Spacey is artistic director.
The two men last worked together on the 1999 film "American Beauty," which won five Academy Awards, including best actor for Spacey and best director for Mendes.
Since then, Mendes - who separated earlier this year from wife Kate Winslet - has directed plays and movies, including "Revolutionary Road" and "Away We Go." Spacey has combined film roles with running the Old Vic.
"Sam is one of the finest directors I have had the pleasure to work with and to gain this opportunity is beyond my wildest hopes," Spacey said.
"With such a remarkable character before me, I have a feeling this one is going to be a memorable experience."
Richard III, who battled for the throne during England's 15th century Wars of the Roses, is one of Shakespeare's great villains and has been played by actors including Laurence Olivier and Ian McKellen.
Mendes said it was "a role I think Kevin is born to play."