Pulsed laser vaporization of graphite generates carbon clusters in a variety of geometries. Injected ion drift tube studies of carbon cluster ions containing 34-64 atoms indicate that it is possible to anneal the non-fullerene isomers into just two dominant geometries: the fullerene and an isomer which appears to be a large monocyclic ring. These results suggest that the monocyclic ring is a relatively stable nonspheroidal form of these all carbon molecules and have important implications for understanding the gas-phase synthesis of fullerenes.