We present experimental evidence that the free-electron laser at the TESLA Test Facility has reached the maximum power gain of 107 in the vacuum ultraviolet (VUV) region at wavelengths between 80 and 120 nm. At saturation the FEL emits short pulses with GW peak power and a high degree of transverse coherence. The radiation pulse length can be adjusted between 30 fs and 100 fs. Radiation spectra and fluctuation properties agree with the theory of high gain, single-pass free-electron lasers starting from shot noise