The DfT has agreed to put £29m towards the full scheme cost of £34.2m
The contract was originally estimated at £42m when the tender race started last year.
Construction information specialist Glenigan said that bidders include Balfour Beatty, Carillion, Lagan, Morgan Sindall and Skanska.
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Transport Minister Baroness Kramer said: “We are putting £29m into this project because the government is serious about investing in the infrastructure the country needs to drive economic growth, both locally and nationally.”
The scheme consists of a new 4.8 kilometre single carriageway road from the A684 north of Bedale to the A684 east of Leeming Bar, which links into Junction 51 of the A1(M).
The A684 is the principal east-west route between Northallerton, the county town of North Yorkshire, and the Yorkshire Dales.
Construction is scheduled to start shortly with completion in May 2016.
