The front of the building is being removed as part of a major revamp, and the restaurant, which was upstairs, will be redeveloped downstairs.
The club had two options - to strengthen or rebuild, and chose the latter. The southern end, facing Burnett Street, will come down.
Upstairs in the new building will be the snooker room, meeting rooms and manager's office.
The MSA was formed in the Ashburton Arcade in 1885 but moved to its present site in the 1920s.
