  We present the results of X-ray spatial and spectral studies of the
``mixed-morphology'' supernova remnant IC 443 using ASCA. IC 443 has a
center-filled image in X-ray band, contrasting with the shell-like appearance
in radio and optical bands. The overall X-ray emission is thermal, not from a
synchrotron nebula. ASCA observed IC 443 three times, covering the whole
remnant. From the image analysis, we found that the softness-ratio map reveals
a shell-like structure. At the same time, its spectra require two (1.0 keV and
0.2 keV) plasma components; the emission of the 0.2 keV plasma is stronger in
the region near the shell than the center. These results can be explained by a
simple model that IC 443 has a hot (1.0 keV) interior surrounded by a cool (0.2
keV) outer shell. From the emission measures, we infer that the 0.2 keV plasma
is denser than the 1.0 keV plasma, suggesting pressure equilibrium between the
two. In addition, we found that the ionization temperature of sulfur, obtained
from H-like K$\alpha$ to He-like K$\alpha$ intensity ratio, is 1.5 keV,
significantly higher than the gas temperature of 1.0 keV suggested from the
continuum spectrum. The same can be concluded for silicon. Neither an
additional, hotter plasma component nor a multi-temperature plasma successfully
accounts for this ratio, and we conclude that the 1.0 keV plasma is
overionized. This is the first time that overionized gas has been detected in a
SNR. For the gas to become overionized in the absence of a photoionizing flux,
it must cool faster than the ions recombine. Thermal conduction from the 1.0
keV plasma to the 0.2 keV one could cause the 1.0 keV plasma to become
overionized, which is plausible within an old (3$\times10^4$ yr) SNR.
