This article solves the Dirichlet problem (existence and uniqueness) for weakly harmonic maps from the unit ball \(B\) into a smooth compact Riemannian manifold \(N\) in a very general boundary situation, in the class of \(H^{1,2}(B,N)\) maps which satisfy a small scaled energy condition. Working on harmonicity in the sense of distributions, the difficulty is not so much the existence of harmonic maps (since energy minimisers can be obtained by the direct method) but finding a suitable class of maps within which the classical problems (here the Dirichlet problem) can be satisfactorily solved. The condition chosen here is the small scaled energy property: \(\sup_{x_{0}\in B, r>0} ( r^{2-n} \int_{B\cap B_{r}(x_{0})} |\nabla u|^{2} dx) \leq \varepsilon^2\) for a sufficiently small \(\varepsilon >0\) (or its boundary version). The existence theorem is proven in two steps, first the case where the boundary data is the restriction of a map from the unit ball into \(N\) and, second, the more general situation where only the boundary of \(B\) is required to map into \(N\). The two situations are linked by an approximation of the boundary data by smooth maps. In both cases the boundary maps and the solutions obtained satisfy the small scaled energy condition. A higher regularity result (i.e. a small scaled energy condition for powers greater than two) is obtained under a similar condition on the boundary data. These results extend from the unit ball to a smooth Riemannian manifold.