Located alongside the existing secondary school, the new 1,953 sq m primary school will have capacity for 420 pupils aged 4 to 11.
They will benefit from modern learning spaces as well as improved sports facilities for both primary and secondary pupils.
In a move away from traditional primary school design, King's Oak will be substantially open plan, offering more flexible teaching space and enabling it to visualise learning in a way not previously possible.
Local highways infrastructure will also be improved as part of the project.
The contractor, which has had offices in Bristol for nearly 50 years, will look to use local suppliers and subcontractors wherever possible.
BAM has a strong track record with South Gloucestershire Council, having built Abbeywood Community School, Severnvale Community Hub and Staple Hill Primary School. It is on site with the final phase of Winterbourne International Academy.
Construction director Neil Dorrington said: "This award not only extends our relationship with South Gloucestershire Council but adds to our considerable success in securing projects through the Construction Framework South West.
"BAM looks forward to using all our skills and experience to create a top class facility for primary pupils in Bristol.
"As a company dedicated to corporate social responsibility, we will engage with local companies and school pupils at every opportunity to ensure that the community gets the most from this important scheme."
BAM is due to start on site in September 2014, with the school expected to be operational in September 2015.
