The Swift Gamma-Ray Burst Explorer performed its first autonomous, X-ray follow-up to a newly detected GRB on 2005 January 17, within 193 s of the burst trigger by the Swift Burst Alert Telescope. While the burst was still in progress, the X-Ray Telescope (XRT) obtained a position and an image for an uncataloged X-ray source simultaneously with the gamma-ray observation. The XRT observed flux during the prompt emission was 1.1 x 10 super(-8) ergs cm super(-2) s super(-1) in the 0.5-10 keV energy band. The emission in the X-ray band decreased by 3 orders of magnitude within 700 s, following the prompt emission. This is found to be consistent with the gamma-ray decay when extrapolated into the XRT energy band. During the following 6.3 hr, the XRT observed the afterglow in an automated sequence for an additional 947 s, until the burst became fully obscured by the Earth limb. A faint, extremely slowly decaying afterglow, a = -0.21, was detected. Finally, a break in the light curve occurred and the flux decayed with a < -1.2. The X-ray position triggered many follow-up observations: no optical afterglow could be confirmed, although a candidate was identified 3" from the XRT position.