Construction work is set to start in the summer.
Five firms were originally shortlisted for the scheme but William Verry dropped out of the race in February.
The 7,000-seat arena, which will sit in the west of the Olympic Park to the south of the hockey centre, will feature external copper cladding.
Glazing will encircle the building at concourse level to allow visitors to watch sport taking place inside, while the roof will have a rainwater harvest feature for toilet flushing.
Olympic Delivery Authority chief executive David Higgins said:“This appointment takes us a step closer towards a simple, sustainable and flexible Handball Arena that works for Games and legacy.”
The Handball Arena will be in the west of the Olympic Park, to the south of the Hockey Centre.
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