Planning permission to renew the centre, which opened in 1971, was granted last year. Work will now start immediately.
The refurbishment design incorporates a new glazed atrium entrance off the High Street.
Queens Square centre manager Melvin Glasby said: “Queens Square has waited a long time for this refurbishment. Following a competitive tender process, A&H stood out for both their experience and commitment to a top specification. They understand our demanding build programme from both the shoppers’ and retailers’ perspectives.”
Queens Square shopping centre will remain open throughout its makeover.