The cohesin complex that mediates sister chromatid cohesion contains three core subunits: Smc1, Smc3, and Scc1. Heterotypic interactions between Smc1 and Smc3 dimerization domains create stable V-shaped Smc1/Smc3 heterodimers with a hinge at the center and nucleotide-binding domains (NBDs) at the ends of each arm. Interconnection of each NBD through their association with the N- and C-terminal domains of Scc1 creates a tripartite ring, within which sister DNAs are thought to be entrapped (the ring model). Crystal structures show that the Smc1/Smc3 hinge has a toroidal shape, with independent "north" and "south" interaction surfaces on an axis of pseudosymmetry. The ring model predicts that sister chromatid cohesion would be lost by transient hinge opening. We find that mutations within either interface weaken heterodimerization of isolated half hinges in vitro but do not greatly compromise formation of cohesin rings in vivo. They do, however, reduce the residence time of cohesin on chromosomes and cause lethal defects in sister chromatid cohesion. This demonstrates that mere formation of rings is insufficient for cohesin function. Stable cohesion requires cohesin rings that cannot easily open. Either the north or south hinge interaction surface is sufficient for the assembly of V-shaped Smc1/Smc3 heterodimers in vivo. Any tendency of Smc proteins with weakened hinges to dissociate will be suppressed by interconnection of their NBDs by Scc1. We suggest that transient hinge dissociation caused by the mutations described here is incompatible with stable sister chromatid cohesion because it permits chromatin fibers to escape from cohesin rings.