BAT and XRT observations of two recent well-covered GRBs observed by Swift, GRB 050315 and GRB 050319, clearly show a prompt component joining the onset of the afterglow emission. By fitting a power-law form to the g-ray spectrum, we extrapolate the time-dependent fluxes measured by BAT, in the energy band 15-350 keV, into the spectral regime observed by XRT, 0.2-10 keV, and examine the functional form of the rate of decay of the two light curves. We find that the BAT and XRT light curves merge to form a unified curve. There is a period of steep decay up to 6300 s, followed by a flatter decay. The duration of the steep decay, 6100 s in the source frame after correcting for cosmological time dilation, agrees roughly with a theoretical estimate for the deceleration time of the relativistic ejecta as it interacts with circumstellar material. For GRB 050315, the steep decay can be characterized by an exponential form, where the e-folding decay time t sub(e) 24 c 2 s (BAT), and t sub(e) 35 c 2 s (XRT). For GRB 050319, a power-law decay -d ln f/d ln t = n, where n 3 provides a reasonable fit. The early-time X-ray fluxes are consistent with representing the lower energy tail of the prompt emission and provide our first quantitative measure of the decay of the prompt g-ray emission over a large dynamic range in flux. The initial steep decay is expected, due to the delayed high-latitude photons from a curved shell of relativistic plasma illuminated only for a short interval. The overall conclusion is that the prompt phase of GRBs remains observable for hundreds of seconds longer than previously thought.