Network Rail confirmed to Construction News Costain had been awarded the design-and-build contract.
Costain has begun work with Network Rail covering the Grip 3 and 4 development phases of the scheme.
Construction of the project is scheduled to start in 2017, with completion expected in 2020.
Costain is currently the lead contractor on the £400m redevelopment of London Bridge, which last month opened its platforms and new concourse.
The work will include doubling the size of the concourse area and the installation of escalators and lifts to improve platform access.
It comes as demand at the station is forecast to increase by a third to 20 million passengers a year by 2025.
A Network Rail spokesman said: “Given the early stages of the development, it’s not possible to put a cost on the scheme, or a value on Costain’s contract.”
In 2014, the value for the station redesign was listed as £120.5m, with the Department for Transport contributing £50m to the scheme, while Network Rail and Gatwick would pay £30m each.