We present evidence for shock acceleration of cosmic rays to high energies (about 10 TeV) in the supernova remnant IC 443. X-ray imaging spectroscopy with ASCA reveals two regions of particularly hard emission: an unresolved source embedded in an extended emission region, and a ridge of emission coincident with the southeastern rim. Both features are located on part of the radio shell where the shock wave is interacting with molecular gas, and together they account for a majority of the emission at 7 keV. Though we would not have noticed it a priori, the unresolved feature is coincident with one resolved by the ROSAT HRI. Because this feature overlaps a unique region of flat radio spectral index (alpha < 0.24), has about equal light-crossing and synchrotron loss times, and a power law spectrum with a spectral index of 2, we conclude that the hard X-ray feature is synchrotron radiation from a site of enhanced particle acceleration. Evidence against a plerion includes a lack of observed periodicity (the pulsed fraction upper limit is 33%), the spectral similarity with the more extended hard region, the location of the source outside the 95% error circle of the nearby EGRET source, the fact that it is nestled in a bend in the molecular cloud ring with which IC 443 is interacting, and the requirement of an extremely high transverse velocity (>5,000 km/s). We conclude that the anomalous feature is most likely tracing enhanced particle acceleration by shocks that are formed as the supernova blast wave impacts the ring of molecular clouds. Comment: 10 pages with embedded figures; accepted by the Ap.J; author's web pages at http://lheawww.gsfc.nasa.gov/users/jonathan/ , http://lheawww.gsfc.nasa.gov/users/rob/Vita/petre.html , http://lheawww.gsfc.nasa.gov/users/evg/evg.html