Cohesin plays a critical role in sister chromatid cohesion, double-stranded DNA break repair and regulation of gene expression. However, the mechanism of how cohesin directly interacts with DNA remains unclear. We report single-molecule experiments analyzing the interaction of the budding yeast cohesin Structural Maintenance of Chromosome (SMC)1-SMC3 heterodimer with naked double-helix DNA. The cohesin heterodimer is able to compact DNA molecules against applied forces of 0.45 pN, via a series of extension steps of a well-defined size ≈130 nm. This reaction does not require ATP, but is dependent on DNA supercoiling: DNA with positive torsional stress is compacted more quickly than negatively supercoiled or nicked DNAs. Un-nicked torsionally relaxed DNA is a poor substrate for the compaction reaction. Experiments with mutant proteins indicate that the dimerization hinge region is crucial to the folding reaction. We conclude that the SMC1-SMC3 heterodimer is able to restructure the DNA double helix into a series of loops, with a preference for positive writhe.