The 273,000 sq ft scheme, now known as the Creechurch Place development, has been in the pipeline for more than 10 years.
The site has been cleared and Skanska said construction will now start in August on the 21-storey project that will be T-shaped in plan.
Skanska is responsible for the construction, mechanical and electrical engineering and installation of services within the main plant rooms.
Ramboll is the structural engineer and Grontmij M&E engineer on the project, which is expected to achieve a BREEAM Excellent rating when completed in September 2016.
The latest deal completes a quartet of big London office schemes for Skanska Construction, including One New Street Square, The Scapel in Lime Street and 1-2 New Ludgate.