The firm saw off bids from Balfour Beatty and Willmott Dixon to take the job, which involves the refurbishment of the existing Grade II listed building.
Construction work will take place across five floors of the landmark 1930s building, and include the creation of a plaza under a double curvature glass canopy.
This will provide extra study space for students from the School of Oriental and African Studies.
The University of London has employed Mace to deliver project management, design and cost consultancy on the project, which will be part of the pilot BREEAM Excellence 2014 award.
Professor Webley said: “The architecturally revolutionary design honours the heritage of the building and caters to the needs of our students, staff and visitors.”
The redevelopment works are due to be completed by January 2016.