Planetary nebulae (PNe) are expected to have a ~10 super(5) K interface layer between the > or =, slanted10 super(6) K inner hot bubble and the ~10 super(4) K optical nebular shell. The PN structure and evolution, and the X-ray emission, depend critically on the efficiency of the mixing of material at this interface layer. However, neither its location nor its spatial extent have ever been determined. Using high-spatial resolution HST STIS spectroscopic observations of the N v ambda lambda1239,1243 lines in the Cat's Eye Nebula (NGC 6543), we have detected this interface layer and determined its location, extent, and physical properties for the first time in a PN. We confirm that this interface layer, as revealed by the spatial distribution of the N v lambda 1239 line emission, is located between the hot bubble and the optical nebular shell. We estimate a thickness of 1.5 x 10 super(16) cm and an electron density of ~200 cm super(-3) for the mixing layer. With a thermal pressure of ~2 x 10 super(-8) dyn cm super(-2), the mixing layer is in pressure equilibrium with the hot bubble and ionized nebular rim of NGC 6543.