The firm beat rivals Bowmer & Kirkland and G F Tomlinson with a £43m bid.
The new car park and bus station development will include a 1,378 space car park, and shops on Carrington Street and Collin Street.
It will be clad in terracotta tile and glass, a radical change from its concrete predecessor.
The project forms part of the wider Broadmarsh shopping centre redevelopment being constructed by Sir Robert McAlpine.
The replacement scheme will include a smarter car park, bus station, space for shops and flats on the Carrington Street elevation.