The offsite facilities for the West End company will see two adjacent buildings converted into one at 1-5 Dryden Street in London.
The refurbishment of the mid 1800s building will give the Donmar new rehearsal facilities, a music room and educational space.
The building will also house a number of administrative offices and a rooftop apartment for use by the company.
Work started this week and is due to be completed for handover in September.
Due to the age of the building 8build will work closely with the Museum of London Archaeology who will monitor the groundworks to the first significant archaeological level with hand excavation.
8build Project Director Carl Raison said: “Our team are keen to get started on this project and create this space for the internationally renowned theatre company; our experience is ever growing in the theatre sector and we look forward to bringing our knowledge to this exciting project.”