  We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains
a properly embedded stable minimal surface, then it is isometric to the
Euclidean space. This implies, for instance, that in presence of a positive ADM
mass any sequence of solutions to the Plateau problem with diverging boundaries
can never have uniform height bounds, even at a single point. An analogous
result holds true up to ambient dimension seven provided polynomial volume
growth on the hypersurface is assumed.
