The decision means Willmott Dixon can start on site this summer with completion due in 2016.
The contract for the glass-fronted Talbot Campus building was awarded through the Scape framework.
The site is currently occupied by a car park and the new building will house research and facilities including seminar rooms and lecture theatres.
The net design and construction cost is expected to be £15.5m, with fit-out likely to handled under a separate contract.
Willmott Dixon expects to spend 62% of the budget with firms within a 40-mile radius of the Talbot Campus during the 18-month building programme.