We present the analysis of between 50 and 100 h of coincident interferometric strain data used to search for and establish an upper limit on a stochastic background of gravitational radiation. These data come from the first LIGO science run, during which all three LIGO interferometers were operated over a 2-week period spanning August and September of 2002. The method of cross correlating the outputs of two interferometers is used for analysis. We describe in detail practical signal processing issues that arise when working with real data, and we establish an observational upper limit on a f−3 power spectrum of gravitational waves. Our 90% confidence limit is Ω0h2100<~23±4.6 in the frequency band 40–314 Hz, where h100 is the Hubble constant in units of 100 km/sec/Mpc and Ω0 is the gravitational wave energy density per logarithmic frequency interval in units of the closure density. This limit is approximately 104 times better than the previous, broadband direct limit using interferometric detectors, and nearly 3 times better than the best narrow-band bar detector limit. As LIGO and other worldwide detectors improve in sensitivity and attain their design goals, the analysis procedures described here should lead to stochastic background sensitivity levels of astrophysical interest. © 2004 The American Physical Society.