Westminster Council has given the go-ahead for the practice's conversion of London's Middlesex Guildhall, which will be a new home for the highest court in the land.
Full planning permission will not be granted until legal agreements are reached with Westminster Council over smaller matters such as car parking, and work will only begin when English Heritage gives listed-building consent.
Problems and controversy have dogged the project since its inception, so news that planning consent has been granted will be a huge boost to Feilden + Mawson.
The firm originally started the scheme with Foster and Partners, which is no longer working on it.
There were numerous reports in the British press that the costs of the project were spiralling out of control, although these were angrily refuted by the Department of Constitutional Affairs (DCA).
The Lord Chancellor, Lord Falconer of Thoroton, said: 'Middlesex Guildhall will be transformed into a building that takes its rightful place alongside its prestigious neighbours on Parliament Square.
'Our plans are heavily influenced by conservation and we believe we have struck the right balance of preserving a historic building while bringing it new life.'