Work on the project, which is to be part-funded by a £2m Sport England lottery grant, is set to begin in February.
Features will include a six-lane 25m competition swimming pool and a learner pool, a studio gym, sauna and steam rooms and an outdoor all-weather sports pitch.
The new centre at Scott Road will replace the former Abbey Leisure Centre, which was destroyed by fire in February last year. Completion of the project is scheduled for January next year.
Designed by Bradshaw Gass and Hope, the leisure centre will comprise energy efficiency measures to reduce running costs, including a combined heat and power plant and LED lighting.
David Price, Business Unit Director, Wates Construction, Yorkshire, said: “We are delighted to have been appointed to deliver the new Abbey Leisure Centre through the YORbuild framework, which is facilitating the cost and time efficient procurement of essential public sector development works.”