An interesting new phenomenon was encountered while evolving a star with a core mass, Mc of 0.8 M0 and with a small envelope mass (0.015 M0) away from the red giant branch towards the nuclei of planetary nebulae, while taking the helium shell flashes into account. It was found that the top of the intershell carbon pocket (the carbon-enriched region in between the hydrogen- and helium-burning shells left behind by the flash) was expanded outwards and cooled immensely; namely, cooled to near 20,000°K! This means that the intershell carbon pocket was lifted out to near the photosphere, right into the shallow outer convective envelope surrounding the hydrogen- and helium-ionization zones! The carbon opacity at these cool temperatures is great. It seems likely that all the layers from the outer regions of the intershell carbon pocket right up to the surface will become convective. This would be a totally new type of deep convective envelope with a vast number of fascinating implications. Careful checks of this new phenomenon are now underway. (Supported in part by the National Aeronautics and Space Administration [NSG 7195].)