Prior to joining the Globe Dromgoole ran the Oxford Stage Company (now Headlong) for seven years. He was artistic director of the Bush Theatre in west London from 1990 to 1996.
His departure date was announced by the Globe today as 23 April 2016, the 400th anniversary of Shakespeare's death.
This will also mark the culmination of Dromgoole's ambitious two-year tour of Hamlet, which aims to visit every country in the world.
He said of the project: "The spirit of touring, and of communicating stories to fresh ears, was always central to Shakespeare's work. We couldn't be happier to be extending that mission even further. By train, coach, plane and boat we aim to take this wonderful, iconic, multifarious play to as many fresh ears as we possibly can."
The production was inspired by last year's Globe to Globe season, during which all 37 Shakespeare plays were staged in a different language by companies from around the world.
The season, which ran as part of the Cultural Olympiad, was a major feather in the cap for Dromgoole's Globe. Other achievements include the construction of the indoor Sam Wanamaker Playhouse, which opens in January, and last year's WhatsOnStage Award-winning revivals of Richard III and Twelfth Night, starring Mark Rylance and Stephen Fry, which transfers to Broadway later this year.