Studies on a resistive switching memory based on a silver-ion-conductive solid polymer electrolyte (SPE) are reported. Simple Ag/SPE/Pt structures containing polyethylene oxide–silver perchlorate complexes exhibit bipolar resistive switching under bias voltage sweeping. The switching behavior depends strongly on the silver perchlorate concentration. From the results of thermal, transport, and electrochemical measurements, it is concluded that the observed switching originates from formation and dissolution of a silver metal filament inside the SPE film caused by electrochemical reactions. This is the first report of an electrochemical “atomic switch” realized using an organic material. The devices also show ON/OFF resistance ratios greater than 105, programming speeds higher than 1 μs, and retention times longer than 1 week. These results suggest that SPE-based electrochemical devices might be suitable for flexible switch and memory applications.