We present the first direct experimental evidence for the charge excess in high-energy particle showers and corresponding radio emission predicted nearly 40 years ago by Askaryan. We directed picosecond pulses of GeV bremsstrahlung photons at the SLAC Final Focus Test Beam into a 3.5 ton silica sand target, producing electromagnetic showers several meters long. A series of antennas spanning 0.3 to 6 GHz detected strong, subnanosecond radio-frequency pulses produced by the showers. Measurements of the polarization, coherence, timing, field strength vs shower depth, and field strength vs frequency are completely consistent with predictions. These measurements thus provide strong support for experiments designed to detect high-energy cosmic rays such as neutrinos via coherent radio emission from their cascades.