The £30m turnover contractor nabbed the job with a £11.5m bid against Interserve’s Yorkshire office, Graham Construction and McLaughlin & Harvey.
The new council offices will be built at Knapping Mount, a brownfield site originally earmarked for housing, which councillors approved planning permission for back in March.
The existing building at Crescent Gardens is being sold to developers to help finance the project, which will see five council offices rationalised into one site.
London architect Farrell & Clark designed the civic building and offices to achieve BREEAM Very Good rating.