We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M× R, where M is asymptotically flat. If the initial hypersurface F⊂ M× R is uniformly spacelike and asymptotic to M× { s} for some s∈ R at infinity, we show that a mean curvature flow starting at F exists for all times and converges uniformly to M× { s} as t→ ∞.