We report experimental results on the diffusivity of water in two ionic liquids obtained using the pulsed-gradient spin-echo NMR method. Both ionic liquids have the same cation, 1-butyl-1-methylpyrrolidinium, but different trifluoromethyl-containing anions. One has a strongly hydrophobic anion, bis(trifluoromethylsulfonyl)amide, while the second has a hydrophilic anion, trifluoromethylsulfonate. Transport of water in these ionic liquids is much faster than would be predicted from hydrodynamic laws, indicating that the neutral water molecules experience a very different friction than the anions and cations at the molecular level. Temperature-dependent viscosities, conductivities, and densities are reported as a function of water concentration to further analyze the properties of the ionic liquid-water mixtures. These results on the properties of water in ionic liquids should be of interest to researchers in diverse areas ranging from separations, solubilizing biomass and energy technologies.