Kirk, who has been in post for four years, will be replaced by the venue's current associate director, Anthony Biggs.
Announcing his departure, Kirk said: "Four years ago the Trustees asked me to take over the artistic helm at Jermyn Street Theatre, and to give it an identity as a producing theatre with a clear artistic output. I feel I have achieved that brief and it is now time for new leadership on the artistic level."
"Anthony Biggs has been named as my successor and I truly cannot think of a better person and a more well-placed individual to take the theatre on to the next level. My heartfelt thanks and respect must go to the incomparable (theatre manager) Penny Horner, who has been my rock at every moment over the past four years. Her stability and passion have made dreams come true."
Kirk has programmed the first six months of 2013, whilst Biggs is concentrating on output from that point on.
Productions from January to July 2013 include a rare revival of Ivor Novello’s Gay’s The Word, which will be followed by Graham Greene’s little-performed play, The Living Room, directed by Tom Littler.
In April, Biggs will direct a revival of Frederick Lonsdale’s 1920s comedy-of-manners, On Approval. Finally the season will culminate with Robert Chevara’s production of In the Bar of a Tokyo Hotel by Tennessee Williams, which has not been seen in London for 30 years.
Currently at the venue, Trevor Nunn's production of Beckett's All That Fall, starring Michael Gambon and Eileen Atkins, continues until 3 November.