We suggest that some of the central stars of planetary nebulae experience a final thermal pulse having achieved a white dwarf configuration and begun their descent along a cooling white dwarf sequence of nearly constant radius. A concrete theoretical calculation demonstrates that, during such a pulse, most of the hydrogen remaining in the star at pulse onset is incorporated into the helium-burning convective shell and completely burned, and that, following the pulse, the star swells briefly to red giant dimensions. The model then proceeds to burn helium on a long time scale, retracing in the H-R diagram approximately the same path that it followed while burning hydrogen during the initial excitation of the nebula, which has by now expanded considerably in extent. We identify as being in the postpulse, quiescent helium burning phase the central stars of the planetary nebulae d Abell 78, and the central stars of a group of related high-excitation objects. These nebulae all have the large radii often found in conjunction with central stars of low luminosity that are thought to be cooling along the white dwarf sequence; however, they have the high luminosities that are characteristic of much smaller nebulae whose nuclei are thought tomore » be proceeding for the first time through the planetary nucleus regime in the (log L, log T/sub e/)-plane.« less