The contract price for the Land Securities job came in at £97m and paves the way for construction to start on the vast office scheme.
It is the third major office win for Skanska with Land Securities and a major coup over its project team over major London rivals Mace and Sir Robert McAlpine.
It will build the 287,000 sq ft office building, which take the form of two connected wedge shape on the site of the International Press Centre on Shoe Lane.
The two flat-iron interlinked buildings will rise to 16 and 14 storeys.
AECOM is on board as mechanical and electrical consultant, with Waterman acting as structural consultant in the Mid-town job.