Gamma-ray bursts (GRBs) are short, intense flashes of 0.1-1 MeV electromagnetic radiation that are routinely observed by Earth orbiting satellites. The sources of GRBs are known to be extragalacitic and located at cosmological distances. Due to the extremely high isotropic equivalent energies of GRBs, which are on the order of Eiso~1054 erg, the gamma-ray emission is believed to be collimated, making them observable only when they are directed towards Earth. The favored progenitor models of GRBs are also believed to emit gravitational waves that would be observable by the current generation of ground-based interferometric gravitational wave detectors. The LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo instruments operated near design sensitivity and collected more than a year of triple coincident data during the S5/VSR1 science run, which spanned the two year interval between November 2005 and October 2007. During this time, GRB detections were being made by the NASA/Goddard Swift Burst Alert Telescope at a rate of approximately 0.3 per day, producing a collection of triggers that has since been used in a coincident GRB-GW burst search with data from the LIGO-Virgo interferometer network. This dissertation describes the search for gravitational waves using the times and locations of 123 below-threshold potential GRB triggers from Swift over the same time period. Although most of the below-threshold triggers are likely false alarms, there is reason to believe that some are the result of actual faintly-observed GRB events. Recent GRB observations indicate that the local rate of low-luminosity GRBs is much higher than previously believed. This result, combined with the possibility of discovering a rare nearby GRB event accompanied by gravitational waves, is what motivates this search. The analysis results indicate no evidence for gravitational waves associated with any of the below-threshold triggers. A median distance lower limit of ~16 Mpc was derived for a typical neutron star-black hole coalescence progenitor assumption.