The firm is understood to have beaten rivals Kier, Thomas Vale and Willmott Dixon with a bid price of £33.3m.
Spread over five floors, the new flagship Faculty of Health and Life Sciences will become home to several departments including sports science, nursing and midwifery, and psychology.
A new two-storey energy building to replace an old combined heat and power plant is also included in the project brief to help achieve a BREEAM excellent rating .
BAM will construct the 120,000 sq ft building, dubbed the super-lab, on the site of the current staff car park in Much Park Street, linking it to an existing sports centre at what will be a shared atrium.
Work is scheduled to start next month with a view to opening in 2017.