We prove that any regular domain in Minkowski space is uniquely foliated by spacelike constant mean curvature (CMC) hypersurfaces. This completes the classification of entire spacelike CMC hypersurfaces in Minkowski space initiated by Choi and Treibergs. As an application, we prove that any entire surface of constant Gaussian curvature in 2+1 dimensions is isometric to a straight convex domain in the hyperbolic plane.