Let $(M,g)$ be a closed smooth Riemannian manifold, and let $(N,h)$ be a compact smooth Riemannian manifold whose sectional curvature satisfies $K_N(\sigma) \leq 0$ for every $2$-plane $\sigma \subset T_qN$ and every $q \in N$. Then for every smooth map $u_0: M \to N$, there exists a smooth harmonic map $u_\infty: M \to N$ such that $u_\infty$ is smoothly homotopic to $u_0$. Equivalently, every smooth homotopy class of maps from $M$ to $N$ contains a smooth harmonic representative.