Molecular dynamics simulations are presented for the self-diffusion coefficient of water in aqueous sodium chloride solutions. At temperatures above the freezing point of pure water, the self-diffusion coefficient is a monotonically decreasing function of salt concentration. Below the freezing point of pure water, however, the self-diffusion coefficient is a non-monotonic function of salt concentration, showing a maximum at approximately one molal salt. This suggests that sodium chloride, which is considered a structure-making salt at room temperature, becomes a structure-breaking salt at low temperatures. A qualitative understanding of this effect can be obtained by considering the effect of ions on the residence time of water molecules near other water molecules. A consideration of the freezing point depression of aqueous sodium chloride solutions suggests that the self-diffusion coefficient of water in supercooled sodium chloride solutions is always higher than that in pure (supercooled) water at the same temperature.