East Ayrshire Council awarded the £8.18m contract to the Co Londonderry contractor after inviting eight contractors to bid.
Construction on the campus, which will bring together St.Xaviers and Patna Primary School is expected to be completed in time for the new school term beginning in August 2012.
It is the second job picked up by the firm under East Ayrshire Council’s “Better School, Better learning proposals.
Earlier this month contractors reacted with anger at the decision by East Ayrshire council to invite 16 contractors to bid for a £7m school job in Scotland.
Three contractors have already dropped out of the race to build Cumnock Primary School after deciding not to waste time drawing up detailed tenders.
The remianing bidders are a mix of Irish, local and national contractors. They are: Ashleigh, Barr Construction, CBC, CCG, Dawn, Graham, Interserve, Kier, Mansell, McLaughlin & Harvey, O’Hare & McGovern, Patten, Pihl