Permission was granted earlier this week for the scheme, first revealed in the AJ (AJ 06.04.06), which is part of a wider masterplan of the Duke of York's barracks in Chelsea.
The go-ahead represents a major victory for Saatchi and his architects, who have had to see off serious and organised local opposition to the project.
The gallery will occupy the entire 4,500m 2building, and will feature a book shop, educational facilities and a café/bar.
The project, commissioned by Charles Saatchi himself, will see the interiors of the Grade II*-listed building pared back to the historic fabric and then concealed behind new walls.
The practices say the aim of the scheme is to make as simple a space as possible and create room for the extraordinary collection to breathe.
This philosophy will be exemplified in the lift shafts, which will be made of simple exposed concrete, and the lighting, which will use Barrislo skins.
Saatchi is being forced to move his collection from its former home in County Hall, on London's South Bank, after he was evicted following a High Court battle with Japanese firm Shirayama Shokusan, the owner of the building.