This Letter reports the results of a search for a stochastic background of gravitational waves (GW) at 100 MHz by laser interferometry. We have developed a GW detector, which is a pair of 75-cm baseline synchronous recycling (resonant recycling) interferometers. Each interferometer has a strain sensitivity of approximately 10;{-16} Hz;{-1/2} at 100 MHz. By cross-correlating the outputs of the two interferometers within 1000 seconds, we found h{100};{2}Omega_{gw}<6 x 10;{25} to be an upper limit on the energy density spectrum of the GW background in a 2-kHz bandwidth around 100 MHz, where a flat spectrum is assumed.