The scheme, which has attracted Heritage Lottery Funding, will see the firm build a three-storey concrete frame structure to link a Regency terrace on one side and the Hugh Casson building, constructed in 1989, on the other.
Designed by Berman Guedes Stretton, the new in-fill building will become the main entrance to Cheltenham Art Gallery and Museum.
It will also house an additional 139 sq m picture gallery, storage and workshop areas, public archive and study areas and 297 sq m of temporary exhibition and gallery areas set over two floors.
As part of the project, ISG will also refurbish the Hugh Casson building, installing new mechanical and electrical services and form link ways into the new extension.
The project is scheduled for completion towards the end of 2012 and the museum and art gallery will re-open to the public by spring 2013.