The winning contractor was tipped by the Enquirer earlier this month.
The original price for the contract in the OJEU notice was estimated at between £27m to £35m last August.
But bidding contractors have struggled to bring it within the original forecast budget.
Designed by Feilden Clegg Bradley Studios the classical music school and performance venue has sparked controversy because of its blocky design and heavy used of brick, needed to give it mass to better acoustic insulation
It will rise to seven stories tall and include a 500-seat concert hall with room for a full orchestra, plus a recital hall with 150 seats and smaller experimental music spaces, organ and jazz rooms.
Hoare Lea & Partners are providing mechanical and electrical design.
Greg Fitzgerald, Galliford Try’s Executive Chairman, said: “We are delighted to be involved in such a high-profile and significant project for both Birmingham and the surrounding area.
“We look forward to working with Birmingham City University and the design team to produce a world-class musical facility for both the students and the wider community.”