The council named the Carlisle civils contractor after the 1.4km Grizebeck improvement scheme gained planning consent allowing for the project to start next year.
Story will now complete the detailed design stage of the two-stage design and build contract with WSP as the design partner.
The new single carriageway road will run between Chapels and Grizebeck on the A595, running to the east of the existing A595, before crossing the existing it and creating a new junction with the A595/A5092 to the west of the existing junction.
Cllr Keith Little, Cumbria County Council Cabinet member for Highways and Transport, said: “Besides improving journey times and boosting our economy, Grizebeck improvements will also create a safer road from Barrow to West Cumbria.”