It will remodel the internal space of the temporary Courtyard Theatre, the original site of the 1974 The Other Place, to create a festival venue for RSC productions and a creative hub.
It will house rehearsals, research and development, and training for artists throughout the year.
Local amateur groups will be able to use the theatre for rehearsals and performances and The Other Place will also be available for commercial hires.
As part of the project, there will also be a new storage facility to house the 30,000 costumes from the RSC's costume store, making them available for public viewing for the first time via theatre tours.
The scheme is part of a wider transformation project that saw the Royal Shakespeare and Swan Theatres re-open in November 2010.
The project is due for completion in the autumn ahead of events in 2016 to mark the 400th anniversary of Shakespeare's death.
Rob Bassett, project manager for BAM whose regional HQ is in Solihull, said: "BAM is delighted to be working on this prestigious facility at the heart of Shakespeare's historic home town.
"We will work hard to ensure the project benefits the local community at construction stage as well as in end use."