In this paper we study sufficient conditions for an asymptotically flat Riemannian manifold to be obtained from a compact Riemannian manifold. For any given asymptotically flat Riemannian manifold whose Weyl and Cotton-York curvature decay faster than $r^{-4}$ and $r^{-5}$, we prove the existence of a distinguished conformal rescaling of the metric which turns the manifold into a compact and regular Riemannian manifold. We also study the case when the decay rate is precisely the critical one.