M&H is understood to have beaten rival Interserve with a bid price of £25.3m for the 100,000 sq ft building.
Site establishment work will start next month, with completion of the building expected in late 2017.
The four to six storey building, designed by Bennetts Architects, is a pivotal piece of the University’s estate and will act as prominent gateway into the University Quarter.
It is the final phase of the Potterrow Development, following the completion of the Informatics Forum and Dugald Stewart Building, and will complete the urban block and courtyard of the original scheme.
The new building will be home to a broad group of occupants from across the University, its start-up and spin-out companies, and industrial collaborators.