The £139m project will link the A589 Morecambe Road, near Torrisholme to a fully remodelled junction 34 on the M6 motorway near Halton.
Around 20 major structures will be built along the 4.8km dual carriageway, including bridges over the West Coast Mainline railway, Lancaster Canal and the River Lune.
Costain will support the council as it takes the scheme through a Public Inquiry, together with developing the design and construction of the scheme.
Andrew Wyllie, chief executive of Costain, said: “We are delighted to have been awarded this ECI contract which further underlines our position as a market leader in the construction and maintenance of highways in the UK.”
It is the second big project Costain has secured with the council. Costain also beat off stiff competition from six rivals to land the £100m M6-Heysham link scheme around Lancaster back in April.