The multisubunit condensin complex is essential for the structural organization of eukaryotic chromosomes during their segregation by the mitotic spindle, but the mechanistic basis for its function is not understood. To address how condensin binds to and structures chromosomes, we have isolated from Saccharomyces cerevisiae cells circular minichromosomes linked to condensin. We find that either linearization of minichromosome DNA or proteolytic opening of the ring-like structure formed through the connection of the two ATPase heads of condensin's structural maintenance of chromosomes (SMC) heterodimer by its kleisin subunit eliminates their association. This suggests that condensin rings encircle chromosomal DNA. We further show that release of condensin from chromosomes by ring opening in dividing cells compromises the partitioning of chromosome regions distal to centromeres. Condensin hence forms topological links within chromatid arms that provide the arms with the structural rigidity necessary for their segregation.