The scheme was originally due to start in 2007 and due to be completed in 2010 but it was pushed back by two years.
The work includes upgrading the two-lane dual carriageway to a three-lane motorway, as well as installing low-noise surfacing to benefit local residents.
Work is now due to finish in 2012.
Paul Smith, chief executive of Morgan Sindall said: "This is a major road project and we have been working on the contract with the Highways Agency since 2004. Morgan Est has an excellent track record in delivering infrastructure projects of this type and we are excited to be working alongside Carillion."
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