Construction of The Defence College of Logistics, Policing & Administration will start this month.
When complete, in summer 2018, it will offer courses focusing on military support services such as catering, transport and military human resources. It will provide living accommodation for up to 2,000 students and staff.
The base currently houses the Adjutant General's Corps but was first used by the Royal Flying Corps in 1917. Winchester City Council granted outline planning permission for its redevelopment last year.
Skanska will also act as development manager for the redevelopment of the Princess Royal Barracks, Deepcut site in Surrey where approximately 1,200 homes, a school and other facilities will be built.
Skanska said that total contract value was approximately £250m.