We have obtained Chandra ACIS-S observations of NGC 6543, the Cat's Eye Nebula. The X-ray emission from NGC 6543 is clearly resolved into a point source at the central star and diffuse emission confined within the central elliptical shell and two extensions along the major axis. Spectral analysis of the diffuse component shows that the abundances of the X-ray-emitting gas are similar to those of the fast (1,750 km/s) stellar wind but not those of the nebula. Furthermore, the temperature of this gas is ~1.7 million K, which is 100 times lower than the expected post-shock temperature of the fast stellar wind. The combination of low temperature and wind abundances is puzzling. The thermal pressure of this hot gas is about twice the pressure in the cool nebular shell; thus, the hot gas plays an essential role in the ongoing evolution of the nebula.