The firm is understood to have beaten Graham Construction, Bam and Sir Robert McAlpine to secure the job with a £10.2m bid.
The four-storey centre will be dedicated to translating research into cures for diseases and will provide laboratories and associated office space.
The internal frame of the building will be all concrete, with a variety of precast elements such as wall panels, columns, stairs, landings and other associated units.
Concast will fabricate, transport and install all precast units on the building.
Antrim Electrical & Mechanical is understood to have won the electrical package with Mott McDonald consulting engineers for the project.