The increased public concerns on healthcare, the environment and sustainable development inspired the development of biodegradable and biocompatible electronics that could be used as degradable electronics in implants. In this work, a fully biodegradable and flexible resistance random access memory (RRAM) was developed with low-cost biomaterial gelatin as the dielectric layer and the biodegradable polymer poly(lactide-coglycolide) acid (PLGA) as the substrate. PLGA can be synthesized by a simple solution process, and the PLGA substrate can be peeled off the handling substrate for operation once the devices are fabricated. The fabricated memory devices exhibited reliable nonvolatile resistive switching characteristics with a long retention time over 104 s and a near-constant on/off resistance ratio of 102 even after 200 bending cycles, showing the promising potential for application in flexible electronics. Degradation of the devices in deionized water and in phosphate buffered saline (PBS) solution showed that the whole devices can be completely degraded in water. The dissolution time of the metals and the gelatin layer was a few days, while that for PLGA is about 6 months, and can be modified by changing the synthesis conditions of the film, thus allowing the development of biodegradable electronics with designed dissolution time.