The construction company will provide the facility as part of a wide-ranging regeneration scheme of the Town Hall Campus for the local authority.
An existing disused technical college will be refurbished and turned into offices including the upgrading of disability access requirements in general to all the existing campus buildings.
BAM worked 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.”
Preliminary works have started, with the main building work to begin in July. The architects are Rick Mather.
Completion is expected in autumn 2013, subject to the impact on the works of the Olympic Games this summer.