Abstract. An a priori Campanato type regularity condition is established for a class of W 1 X local minimisers u of the general variational integralwhere Ω ⊂ R n is an open bounded domain, F is of class C 2 , F is strongly quasi-convex and satisfies the growth conditionfor a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato spaceand the space of bounded mean oscillation BMO(Ω, R N×n ). The admissible maps u : Ω → R N are of Sobolev class W 1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W 1 BMO local minimisers is extended from Lipschitz maps to this admissible class.