Michael Grandage directs current Donmar hit Privates on Parade
Mr Grandage, 39, will replace the Oscar-winning director when he steps down in November.
The former actor has directed a string of successes at the 251-seat Donmar including the current hit Privates on Parade.
Mendes is leaving the venue in Covent Garden, central London to pursue a freelance film and theatre career.
Big-seller
Mr Grandage will take on the new role while keeping his commitment to a three-theatre complex in Sheffield.
On 19 March, he opens a revival of Shakespeare's Richard III in the Yorkshire city with Kenneth Branagh in the title role.
The production has been the fastest-selling show in the 31-year history of the Crucible Theatre.
He also persuaded Shakespeare in Love star Joseph Fiennes to return back to the English stage in Edward II.
Mr Grandage started his life in front of the screen as an actor in TV shows including BBC One's sci-fi drama Bugs and ITV1's Cadfael.
He also starred in the Oscar-winning drama The Madness of King George.
However in a change of career he moved into theatre directing and with considerable success.
Last year, his Donmar première of Stephen Sondheim's Merrily We Roll Along won three Olivier Awards, including best musical.
Revamp
Mendes took charge of the Donmar in 1992, when the theatre was a struggling West End fringe venue.
He revamped its reputation with productions of stage classics such as The Glass Menagerie, Company, Cabaret and Habeas Corpus.
One of his major coups was to put Nicole Kidman on stage in New York and the Donmar where she appeared - scantily clad - in The Blue Room.
The production made national headlines, with one newspaper describing the performance as "theatrical Viagra".
Mendes was offered the job of directing the movie American Beauty after Steven Spielberg was impressed by his productions of Oliver and Cabaret.
The film won five Academy Awards, including best director for Mendes.
He will release his next movie later this year - The Road to Perdition stars Tom Hanks, Jude Law and Paul Newman.