The company was appointed under the Construction Framework South West.
The centre is to undergo two-phase re-modelling of its existing learning and roof terrace areas.
BAM will develop a new 1200m2 extension in the west of the gallery, below the Barnoon car park.
The company will also create a gallery space for temporary exhibitions along with improving art handling and additional visitor facilities.
The new building will be linked with the existing gallery to allow visitors to move from one to the other.
BAM commenced enabling works in October 2014 and has been discussing the detailed design of the main project works which is scheduled to commence in June.
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Graham Kingdon, BAM's Construction Director, said: "This is a stunning building. Not only does it require the highest standards and finishes to protect and enhance the display of priceless artworks but it is also logistically very challenging because of the access constraints.
"We’ll be working with BAM’s specialist services engineering company which now covers the West of England and Wales, and the existing gallery of the Tate Gallery will be open for much of the build, closing only for structural works during the winter."
BAM has already removed 1000m3 of blue Elvan rock, conducted temporary works and extended the retaining walls.
The new building will have exposed concrete finishes and granite cladding.
Two architects, including the original designers of the gallery, are involved in the project, Jamie Fobert and Evans and Shalev.
Structural engineering will be conducted by Price and Myers, with mechanical and electrical services being provided by Max Fordham.
Project management and quantity surveying will be carried out by Sweett Group.
Work is scheduled to be completed in Spring 2017.
(LM/CD)
