We present results of computer simulations of the mobilities of the alkali metal ions (Li+, Na+, K+, Rb+, and Cs+) and the halides (F-, Cl-, Br-, and I-) at 25 degrees C using the SPC/E model for water and ion-water parameters fitted to the binding energies of small clusters of ions. A simple truncation of the ion-water and water-water potentials was used, and the mobilities calculated from the mean square displacement and the velocity autocorrelation functions, respectively, were found to be in good agreement with each other. The calculations demonstrate, for the first time, cation and anion mobilities that fall on separate curves, as functions of ion size, with distinct maxima. This is in complete accord with experimental trends observed in water at 25 degrees C. The cation mobilities are also in better agreement with the measured values than the calculations done earlier (J. Chem. Phys. 1994, 101, 6964) using the TIP4P model. The mobilities of the halides calculated here for the SPC/E model are however slightly lower than the experimental results. The residence times of water in the hydration shells around an ion are found to decrease dramatically with its size. Stereoscopic pictures show that the structure of the solvent cage around an ion is qualitatively different for the larger ions, implicating both solvent dynamics and structure as important factors in explaining ion mobility in aqueous systems.