A stochastic background of gravitational waves is expected to arise from a superposition of many incoherent sources of gravitational waves, of either cosmological or astrophysical origin. This background is a target for the current generation of ground-based detectors. In this article we present the first joint search for a stochastic background using data from the LIGO and Virgo interferometers. In a frequency band of 600-1000 Hz, we obtained a 95% upper limit on the amplitude of Omega(GW)(f) = Omega(3)(f/900 Hz)(3), of Omega(3) < 0.32, assuming a value of the Hubble parameter of h(100) = 0.71. These new limits are a factor of seven better than the previous best in this frequency band.