The firm has been locked in exclusive negotiations for months over the project but is now understood to have secured the job after tough negotiations for a fixed price of £130m.
Demolition work began at the end of last year on the project, which is part of the £2bn London Bridge Quarter development.
The 17-storey building is designed as the companion building to the £425m Shard of Glass tower, which Mace is also building.
Renzo Piano designed the 420,000 sq ft sister building, which will boast a 11,000 sq ft solar panel roof, helping it to achieve a 5 star BREEAM rating.
