The transport of water and ions through membranes formed from carbon nanotubes ranging in diameter from 6 to 11 A is studied using molecular dynamics simulations under hydrostatic pressure and equilibrium conditions. Membranes incorporating carbon nanotubes are found to be promising candidates for water desalination using reverse osmosis, and the size and uniformity of tubes that is required to achieve a desired salt rejection is determined. By calculating the potential of mean force for ion and water translocation, we show that ions face a large energy barrier and will not pas' through the narrower tubes studied ((5,5) and (6,6) "armchair" type tubes) but can pass through the wider (7,7) and (8,8) nanotubes. Water, however, faces no such impediment due to the formation of stable hydrogen bonds and crosses all of the tubes studied at very large rates. By measuring this conduction rate under a hydrostatic pressure difference, we show that membranes incorporating carbon nanotubes can, in principle, achieve a high degree of desalination at flow rates far in excess of existing membranes.