The firm beat rivals Interserve, Wates and Bardsley to take the building job, which is expected to start on site early next year.
Aedas’ designs include three swimming pools, two multi-court sports halls, squash courts a climbing wall and bouldering room.
Tesco will take over the site of the old sport centre once it has been demolished to build a new supermarket and make a contribution to the new building in Springwood.