The switching speed of a Cu/Ta2O5/Pt atomic switch between a high-resistance (OFF) state and a low-resistance (ON) state was evaluated by transient current measurements under the application of a short voltage pulse. It was found that the SET time from the OFF state to the ON state decreased as low as 1 ns, and the RESET time from the ON state to the OFF state reached a few ns using moderate pulse amplitudes. The switching time depends strongly on the pulse amplitude and the cell resistance before applying a voltage pulse. This observation indicates that oxide-based atomic switches hold potential for fast-switching memory applications. It was also found that Cu nucleation on the Pt electrode is likely to the rate-limiting process determining the SET time and the REST time appears to be preferentially determined by thermochemical reaction.