ISG will build an £80M, 6,000 seater permanent structure designed by Hopkins Architects and consultant Expedition Engineering.
The contractor's previous experience of building cycling venues extends to the replacing of track at the Manchester Velodrome ahead of the recent World Cycling Championships.
"To be working on such a high-profile project is hugely exciting, particularly after the recent success of the British Cycling Team," said ISG managing director Steve Trotter.
"We couldn't have hoped for a more exciting project to follow on from our work on the Manchester Velodrome."
Olympic Delivery Authority chairman David Higgins said: "The VeloPark will be one of the biggest legacies of the Games, delivering world-class cycling facilities for elite athletes and the local community to use for generations to come, all in a landmark new building for London."