A two-dimensional (2D) porous layer can make an ideal membrane for separation of chemical mixtures because its infinitesimal thickness promises ultimate permeation. Graphene--with great mechanical strength, chemical stability, and inherent impermeability--offers a unique 2D system with which to realize this membrane and study the mass transport, if perforated precisely. We report highly efficient mass transfer across physically perforated double-layer graphene, having up to a few million pores with narrowly distributed diameters between less than 10 nanometers and 1 micrometer. The measured transport rates are in agreement with predictions of 2D transport theories. Attributed to its atomic thicknesses, these porous graphene membranes show permeances of gas, liquid, and water vapor far in excess of those shown by finite-thickness membranes, highlighting the ultimate permeation these 2D membranes can provide.