The job is part of a wide-ranging regeneration scheme of the Town Hall Campus for the local authority.
The Grade II listed town hall buildings will also be refurbished under the East Ham Civic Campus scheme which has a total value of £20.8m
Main construction work will start next month and the scheme is due for completion in autumn 2013.
BAM worked closely with the councils design team and English Heritage to get planning approval for the alteration work to the listed buildings.
Construction director Andy Mason said: “BAM is very pleased to be delivering these important facilities for Newham Council.
“A great deal of work was required from all parties to make this scheme a reality, and BAM is delighted to have played its part in making that happen.
“We look forward to creating some excellent facilities for council staff and local residents.”
The architects are Rick Mather, who worked with BAM on the Ashmolean Museum of Art and Archaeology in Oxfordshire.