We investigate the permeability and selectivity of graphene sheets with designed subnanometer pores using first principles density functional theory calculations. We find high selectivity on the order of 10(8) for H(2)/CH(4) with a high H(2) permeance for a nitrogen-functionalized pore. We find extremely high selectivity on the order of 10(23) for H(2)/CH(4) for an all-hydrogen passivated pore whose small width (at 2.5 A) presents a formidable barrier (1.6 eV) for CH(4) but easily surmountable for H(2) (0.22 eV). These results suggest that these pores are far superior to traditional polymer and silica membranes, where bulk solubility and diffusivity dominate the transport of gas molecules through the material. Recent experimental investigations, using either electron beams or bottom-up synthesis to create pores in graphene, suggest that it may be possible to employ such techniques to engineer variable-sized, graphene nanopores to tune selectivity and molecular diffusivity. Hence, we propose using porous graphene sheets as one-atom-thin, highly efficient, and highly selective membranes for gas separation. Such a pore could have widespread impact on numerous energy and technological applications; including carbon sequestration, fuel cells, and gas sensors.