Until recently, X-ray flares during the afterglow of gamma-ray bursts (GRBs) were a rarely detected phenomenon; thus, their nature is unclear. During the afterglow of GRB 050502B, the largest X-ray flare ever recorded rose rapidly above the afterglow light curve detected by the Swift X-Ray Telescope. The peak flux of the flare was >500 times that of the underlying afterglow, and it occurred >12 minutes after the nominal prompt burst emission. The fluence of this X-ray flare, (1.0 c 0.05) x 10 super(-6) ergs cm super(-2) in the 0.2-10.0 keV energy band, exceeded the fluence of the nominal prompt burst. The spectra during the flare were significantly harder than those measured before and after the flare. Later in time, there were additional flux increases detected above the underlying afterglow, as well as a break in the afterglow light curve. All evidence presented below, including spectral and, particularly, timing information during and around the giant flare, suggests that this giant flare was the result of internal dissipation of energy due to late central engine activity, rather than an afterglow-related effect. We also find that the data are consistent with a second central engine activity episode, in which the ejecta is moving slower than that of the initial episode, causing the giant flare and then proceeding to overtake and refresh the afterglow shock, thus causing additional activity at even later times in the light curve.