The new 4-mile dual carriageway will run between Junction 19 of the M6 near Knutsford and Junction 7 of the M56 near Bowdon.
It will take traffic away from local villages, by-passing Over Tabley, Mere and Bucklow Hill.
The new route will run to the west of the existing A556 and will be built below ground level where possible to minimise the impact on the surrounding area.
The road will rejoin the existing A556 for a short section near Bucklow Hill, which will be upgraded.
Improvements will also be made to the layout of Junction 7 of the M56.
Costain’s £101.7m contract award follows on from the firm’s delivery of the Early Contractor Involvement phase of the scheme.
Work will commence immediately, with the new link road expected to open in 2016/7.
Andrew Wyllie, chief executive of Costain, said: “This award is further demonstration of the importance to Costain of long-term, strategic partnerships with our customers, and our ability to deliver a range of innovative integrated services.”