The contract is worth £115m – substantially less than the original £150m construction cost estimate.
The Enquirer revealed earlier this month that Kier was in line for the job.
Work will involve construction of a new hospital, containing 162 mental health beds, to replace the facilities currently provided in outdated buildings.
Construction on the hospital is expected to start late in 2013 and phase 1 will be ready to accept patients by January 2017.
Other bidders in the race included Balfour Beatty, Skanska, Vinci UK and Willmott Dixon.
It is understood that Kier’s bid was up to £18m lower than the next cheapest.