We present version 1.0 of the NASA Galaxy Evolution Explorer (GALEX) ultraviolet variability (GUVV) catalog, which contains information on 84 time-variable and transient sources gained with simultaneous near-ultraviolet (NUV) and far-ultraviolet (FUV) photometric observations. These time-variable sources were serendipitously revealed in the various 1.2d diameter star fields currently being surveyed by the GALEX satellite in two ultraviolet bands (NUV 1750-2750 Å, FUV 1350-1750 Å) with limiting AB magnitudes of 23-25. The largest amplitude variable objects currently detected by GALEX are M dwarf flare stars, which can brighten by 5-10 mag in both the NUV and FUV bands during short-duration (<500 s) outbursts. Other types of large-amplitude ultraviolet variable objects include ab-type RR Lyrae stars, which can vary periodically by 2-5 mag in the GALEX FUV band. This first GUVV catalog lists galactic positions and possible source identifications in order to provide the astronomical community with a list of time-variable objects that can now be repeatedly observed at other wavelengths. We expect the total number of time-variable source detections to increase as the GALEX mission progresses, such that later version numbers of the GUVV catalog will contain substantially more variable sources.