We describe studies of the annealing and dissociation of even-numbered carbon cluster ions containing 50-70 atoms. The polycyclic polyyne ring isomers for cluster ions in this size regime can be annealed almost entirely into the fullerene geometry, with a small fraction being converted into an isomer which appears to be a large monocyclic ring. Suprisingly, we find that C-60(+) behaves essentially the same as other similarly-sized clusters (such as C-58(+)). This has important implications for understanding the mechanism of these structural transformations, as well as the overall scheme for fullerene synthesis. The activation energies for conversion of the polycyclic rings to fullerenes are low and relatively independent of cluster size, though the efficiency of forming a fullerene (rather than a large monocyclic ring) increases with cluster size. Based on our experimental results, a detailed mechanism is proposed to account for conversion of the polycyclic polyyne rings into fullerenes. According to this mechanism, a fullerene fragment is prepared by a Bergman enediyne cyclization followed by a radical-induced ring closure and a retro [2 + 2] process. The polyyne chains are then configured to spiral around the fullerene fragment and zip up to form a spheroidal fullerene. We also consider how these processes fit into an overall scheme for fullerene synthesis from small carbon fragments and describe a scheme that is consistent with the experimental results presented here.