Willmott is believed to have fought off competition from Bam, Vinci and Graham Construction to land the deal.
Work will start next month to provide a new home for the 170-pupil school with a new library, IT spaces, therapy rooms, hydro-pool, sports hall and vocational teaching areas.
Existing residential facilities will also be extensively refurbished to provide self contained flats with a new entrance offering a clear division between school and home.
The final phase will include improved vehicle access and parking, new grassed and multi-use sports pitches together with new habitat and recreation areas.
Anthony Dillon, managing director for Willmott Dixon in the region, said: “We’ll be making sure that local business are used as much as possible under our Local Pound strategy that ensures local people benefit from this substantial investment by Conwy County Borough Council.
“We have built special needs schools across the country and will be applying all that experience here to deliver a superb new school for the region that provides very specialised learning.”
Jonathan Morgan, Headteacher, Ysgol Y Gogarth said: “The pupils, staff and Governors are extremely excited that work is about to start on the new building.
“The additional specialist resources, teaching facilities and technology will give us a building that matches the expertise of staff and will enable us to meet our high expectations for pupils’ potential.
Work will take place in phases over two years, with Willmott Dixon working with Atkins as architect.