Westminster City Council has approved the practice's proposals, which will replace temporary accommodation off Parliament Square, after overcoming a number of initial reservations about the project.
Council officials had originally raised concerns in an October meeting over the scheme's disabled access and design - including the choice of materials and colour, as well as its impact on the Palace of Westminster World Heritage site.
ADP was sent back to the drawing board to reconsider the design of a reception centre and information kiosk which would accommodate the needs of those visiting the Commons.
Despite these initial reservations, councillors have voted through the project after being satisfied that the architect had dealt with their concerns.
An ADP spokesperson confirmed the practice's relief, and added: 'We're getting there.'