The contractor will redevelop the Murrays' Mills area into 124 one, two and three-bedroom apartments for the Manchester Life Development Company.
The scheme is part of Manchester Life's plans to build over 900 homes in Ancoats and New Islington.
Planning permission and listed building consent, to re-use the surviving steam powered cotton mills, was granted in July 2015.
Work is due to begin this month, with the apartments ready for occupation by autumn 2017.
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Marty Edelman, Executive Chairman of Manchester Life Development Company, said: "Our absolute priority is to preserve the heritage of one of Manchester's most important historical buildings, and in doing so, create a flagship residential scheme in keeping with the character of Ancoats. Murrays' Mills will provide a unique living experience for the 124 owners, and importantly will serve as a catalyst for further regeneration in the area."
Gary Hughes, Northern Regional Director at Graham, added: "GRAHAM is delighted to have been entrusted with the transformation of Murrays' Mills into what will be a first class and unique residential scheme.
"We have worked closely with Manchester Life and its design partners over the past year to ensure that every aspect of the restorative conversion of Murrays' Mills protects the building and fulfils the vision for the Manchester Life living experience."
(LM)
