Diffusion of tracer dye molecules in water confined to the nanoscale is an important subject with a direct bearing on many technological applications. It is not yet clear, however, if the dynamics of water in hydrophilic as well as hydrophobic nanochannels remains bulk-like. Here, we present diffusion measurement of a fluorescent dye molecule in water confined to the nanoscale between two hydrophilic surfaces whose separation can be controlled with a precision of less than a nm. We observe that the fluorescence intensities correlate over fast (∼30 μs) and slow (∼1000 μs) time components. The slow time scale is due to adsorption of fluorophores to the confining walls, and it disappears in the presence of 1 M salt. The fast component is attributed to diffusion of dye molecules in the gap. It is found to be bulk-like for sub-10 nm separations and indicates that the viscosity of water under confinement remains unaltered up to a confinement gap as small as ∼5 nm. Our findings contradict some of the recent measurements of diffusion under nanoconfinement; however, they are consistent with many estimates of self-diffusion using molecular dynamics simulations and measurements using neutron scattering experiments.