The consortium, called Project Blue, is being led by Qatari Diar - a property investor from Qatar - and CPC Group, a Guernsey-based investment company run by Christian Candy.
Originally built to house two battalions of troops, the barracks are now home to the Public Duties and State Ceremonial personnel.
The scheme will prove a massive opportunity for both Rogers and the Candy brothers, who saw off some impressive bids - including a team featuring architect Michel Mossessian - to land the prize project.
It will be the second time that Rogers will work with the Candy brothers: the recent Pritzker Prize winner designed One Hyde Park - currently London's most expensive flats - which the brothers are building through their other development company, Candy & Candy.
Qatari Diar chief executive Nasser Al-Ansari said: 'We are delighted to lead the consortium which has purchased this site. This project is very important to us as it represents our first major investment in Europe.'