Let M be a complete, connected, two-dimensional Riemannian manifold with nonpositive Gaussian curvature K. We say that M satisfies the unrestricted complete controllability (UCC) property for the Dubins problem if the following holds given any (p(1), v(1)) and (p(2), v(2)) in TM, there exists, for every epsilon > 0, a curve gamma in M, with geodesic curvature smaller than epsilon, such that. connects p1 to p2 and, for i = 1, 2, gamma is equal to v(i) at p(i). Property UCC is equivalent to the complete controllability of a family of control systems of Dubins' type, parameterized by epsilon. It is well known that the Poincare half-plane does not verify property UCC. In this paper, we provide a complete characterization of the two-dimensional nonpositively curved manifolds M, with either uniformly negative or bounded curvature, that satisfy property UCC. More precisely, if sup(M) K < 0 or inf(M) K > -infinity, we show that UCC holds if and only if (i) M is of the first kind or (ii) the curvature satisfies a suitable integral decay condition at infinity.