The contractor will build 5.2km of bypass and associated junctions and employ around 165 workers during the months of peak activity.
Will Bros beat rivals Farrans Roadbridge, John Paul Construction and R J McLeod to take the award which was let four months later than first timetabled.
Once complete the scheme will separate local road users from those travelling longer distances, including to the key ports of Stranraer and Cairnryan.
Michael Matheson, Cabinet Secretary for Transport, Infrastructure and Connectivity said: “The local community will benefit from opportunities created for small and medium enterprises to bid for subcontractor roles and the provision of at least four vocational and seven professional, site-based training opportunities.”