The value of Galliard's contract is approximately £100m.
The six-storey Edwardian redbrick building, built in 1904-1906, is being converted into 263 flats and 108 serviced apartments.
Called Islington Square, the developer is Sager Group and architect is CZWG.
First residents are expected to move in by early 2017 and final completion is planned for the end of 2017.
Sager Group CEO Giris Rabinovitch said: “Islington Square is one of the most prestigious regeneration projects in London, and will provide an exciting new Covent Garden style destination for the capital."