Self-amplified spontaneous emission in a free-electron laser has been proposed for the generation of very high brightness coherent x-rays. This process involves passing a high-energy, high-charge, short-pulse, low-energy-spread, and low-emittance electron beam through the periodic magnetic field of a long series of high-quality undulator magnets. The radiation produced grows exponentially in intensity until it reaches a saturation point. We report on the demonstration of self-amplified spontaneous emission gain, exponential growth, and saturation at visible (530 nanometers) and ultraviolet (385 nanometers) wavelengths. Good agreement between theory and simulation indicates that scaling to much shorter wavelengths may be possible. These results confirm the physics behind the self-amplified spontaneous emission process and forward the development of an operational x-ray free-electron laser.