Under the terms of the contract, which is worth £400m over five years, Costain will carry out the detailed design and delivery of the project.
Work is due to commence in full in 2013 and is scheduled for completion in 2018.
The Network Rail deal is a big win for Costain and builds on its £80m Reading railway station upgrade job awarded in July.
Costain managed to nose ahead of shortlisted rival Skanska to take the job after they saw off bids from Laing O’Rourke/Mace, Balfour Beatty and Carillion in the earlier tendering rounds.
Costain will ensure that the station remains fully operational at all times to minimise disruption to passengers.
It will involve the provision of a new concourse at street level, with entrances on Tooley Street and St Thomas Street, which will increase passenger capacity through the station from 50m to 75m.
Entrance and exit points will also be redeveloped to improve ease of access to and from the station.
The redevelopment of London Bridge Station is a key part of the Thameslink programme, which aims to increase significantly the total frequency of trains through the station per hour.
Andrew Wyllie, chief executive of Costain, said: “We are looking forward to commencing work on this very important project, which reinforces our position as one of the UK’s leading engineering solutions providers.
“Our focus is to design and deliver a world class London Bridge Station which enhances the capital’s infrastructure and improves the journeys of the hundreds of thousands of passengers who pass through the station every day.”