The half a billion pound, six-year contract has been awarded to Ferrovial Agroman Laing O’Rourke and covers the main construction works.
The full cost of the scheme, expected to be up to £1bn, is being funded entirely through developer contributions, including Battersea Power Station and the new Enterprise Zone in Nine Elms.
Construction is expected to start next year. Two new Tube stations will be built - one at Battersea Power Station and another just off Wandsworth Road.
They will connect to the rest of the Northern Line at Kennington Station and could be open by 2020.
The people of Battersea will get access to the Tube network for the first time and journey times to the West End and the City will be cut to just less than 15 minutes.
Mike Brown MVO, the managing director of London Underground, said: "This is an important step in ensuring the delivery of this vital extension which will significantly improve transport links in the area."
Wandsworth Council leader Councillor Ravi Govindia said: "It’s fantastic news that a contractor has been appointed and we are a step closer to achieving this vital Tube link that we have been campaigning for so long. It will breathe new life into Nine Elms and yield billions of pounds in new economic growth and tax revenue."