The project was first estimated at around £31m two years ago but has now risen to an expected £35m.
Volker will deliver a 1.7km long single carriageway road that includes a 500m-long viaduct crossing of the river Great Ouse and a separate bridge over a railway line.
The new route leaves the A142 Stuntney Causeway to the south-east of Ely, before re-joining the existing A142 at a new roundabout on Angel Drove.
Detailed design work on the two-year contract will now get underway to start on site at the beginning of 2017.
As part of the new road bridge over the River Great Ouse there will be a walkway attached to the bridge that will face towards Ely. The walkway will link the Fen Rivers Way and Ouse Valley Way footpaths together, providing a new circular walking route for residents and visitors to Ely.
It will also improve access to the River Great Ouse Country Wildlife Site and give new views of Ely and its surrounding landscape.
Tony Gee is the structural consultant and Trevor Waude & Co the QS on the job.