The firms beat off rivals Morgan Sindall, Shepherd, Graham and R G Carter with a bid of £15.4m.
The scheme comprises redevelopment of Perinatal Services at Addenbrookes Hospital in order to provide a 7,200 sq m 3-storey extension to the Rosie maternity unit.
This is being expanded to cope with projected demand in maternity deliveries and neonatal activity to 2021.
When the Rosie Hospital was built in 1983, it was built for 3,000 to 4,000 births a year. The unit now has around 5,800 births a year and this is predicted to go up to 7,500 in the next few years.
The associated works include enabling, sewer systems, infrastructure and landscaping.