Their redesign and refurbishment of the Grade II*-listed building on Parliament Square will see it transformed by late 2008 into the home of the highest court in the land.
The details of the project will be subject to the approval of Westminster City Council and English Heritage.
Middlesex Guildhall's existing Crown Court work will be relocated elsewhere in London, and plans have been proposed to ensure there will be no adverse effect on the criminal justice system as a result of these changes.
The secretary of state for constitutional affairs and lord chancellor Lord Falconer said he was delighted with the decision.
'We have assembled an absolutely first-class team to see through the development of what is destined to become one of this country's landmark buildings,' he said.
'All those involved in the successful bid have track records of excellence. Their brief is to come up with a building design in keeping with the architectural heritage of the area, but to the highest modern standards.
'By 2008 we should have a Supreme Court of which we can all be proud - open and accessible to all, with excellent accommodation for judges and support staff alike,' he added.