Bursts of coherent synchrotron radiation at far-infrared and millimeter wavelengths have been observed at several storage rings. A microbunching instability has been proposed as the source for the bursts. However, the microbunching mechanism has yet to be elucidated. We provide the first evidence that the bursts are due to a microbunching instability driven by the emission of synchrotron radiation in the bunch. Observations made at the Advanced Light Source are consistent with the values predicted by the proposed microbunching model. These results demonstrate a new instability regime for high energy synchrotron radiation sources and could impact the design of future sources.