The Trust has signed an eight-year lease for the mansion. It is keen to house its 60 staff in the new office.
After the BBC London and BBC Training offices in Marylebone High Street were sold off in October, the Trust rejected an accommodation within the existing BBC buildings and wanted a "physical separation" to protect its independence.
The Trust's action has invited severe criticism from the shadow culture secretary, Jeremy Hunt, and the Liberal Democrat culture spokesman, Don Foster.
Meanwhile, the redevelopment of the BBC's Broadcasting House in the West End, which was supposed to cost £1bn, is expected to cross the budget by £59m.