The 138-room post-graduate halls development is in the city’s Southside conservation area at Buccleuch Place and Meadow Lane.
Graham will develop the former garages and workshops site with a series of mixed-size buildings to tune in with the surrounding area.
This includes a new 5-storey block, offering a contemporary take on the City’s traditional tenement design, which will be constructed using natural stone, glass and zinc.
The scheme has been designed by JM Architects while RSP Consulting is the building service engineer.
It is Graham’s second major success with the University in the last month. The firm was also named as the successful bidder for Edinburgh’s £20m Old College refurbishment project.
This involves refurbishing the iconic 1789 Law School Old College in the city.