We report the results of Swiftobservations of the Gamma Ray Burst GRB 050603. With a V magnitude V=18.2 10 hours after the burst, the optical afterglow was the brightest so far detected by Swift and one of the brightest afterglows ever seen at late times. The Burst Alert Telescope (BAT) trigger light curves showed three fast-rise-exponential-decay spikes with T 90=12±2s and a fluence of 7.6 × 10-6 ergs cm-2 in the 15-150 keV band. The Swift spacecraft started the observations of the afterglow with the narrow-field instruments about 10 hours after the detection of the burst. The burst was bright enough to be detected by the Swift UV/Optical telescope (UVOT) for almost 3 days and by the X-ray Telescope (XRT) for a week after the burst. The X-ray light curve showed a rapidly fading afterglow with a decay index α=1.76-0.07 +0.15 This is one of the steepest late time slopes seen by Swift so far. The X-ray energy spectral index was βX=0.71±0.10 with the absorption parameter in agreement with the Galactic value. The spectral analysis did not show an obvious change in the X-ray spectral slope over time. The optical UVOT light curve decayed with a slope of 1.97. The steepness and the similarity of the optical and X-ray decay rates suggest that the afterglow was observed after the jet break. Its redshift z=2.821 was measured by Berger & Becker (2005).