BAM fought off competition from Carillion, Colas, Dean & Dyball and Osborne to work on the scheme, which received planning permission last month.
The works involve the construction of a 4.7km single carriageway road from Fareham to Gosport. They also include installation and diversion of services, a sustainable water surface drainage network and bus shelters with cycle storage.
The project is set to take two years to complete.
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