We have observed "giant magnetoresistance'' in short-period Cu/Co-Ni-Cu alloy superlattices electrodeposited from a single electrolyte under potentiostatic control. The superlattices were grown on polycrystalline Cu substrates which were removed before transport measurements were made. Room-temperature magnetoresistances of over 15% in applied magnetic fields of up to 8 kOe were observed in superlattices having Cu layer thickness of less than 10 angstrom.