The contractor, which has offices in Gateshead, will create a three-storey, U-shaped building for the Teesside institution.
BAM construction director Andrew Warwick said the firm would be aiming to start on site in March 2015, subject to the planning process.
When finished in Spring 2016, the building will cater for around 750 pupils as well as be widely used by the Ingleby Barwick community, with sports, café and computer facilities among those to be made available in the evenings and at weekends.
School pupils will be able to use the construction project as a learning resource, and BAM will seek to work with local suppliers where practical.
Sustainability features in the new school building will include solar panels on the roof and natural ventilation.