Knowledge of the elastic properties of DNA is required to understand the structural dynamics of cellular processes such as replication and transcription. Measurements of force and extension on single molecules of DNA have allowed direct determination of the molecule's mechanical properties, provided rigorous tests of theories of polymer elasticity, revealed unforeseen structural transitions induced by mechanical stresses, and established an experimental and conceptual framework for mechanical assays of enzymes that act on DNA. However, a complete description of DNA mechanics must also consider the effects of torque, a quantity that has hitherto not been directly measured in micromanipulation experiments. We have measured torque as a function of twist for stretched DNA--torsional strain in over- or underwound molecules was used to power the rotation of submicrometre beads serving as calibrated loads. Here we report tests of the linearity of DNA's twist elasticity, direct measurements of the torsional modulus (finding a value approximately 40% higher than generally accepted), characterization of torque-induced structural transitions, and the establishment of a framework for future assays of torque and twist generation by DNA-dependent enzymes. We also show that cooperative structural transitions in DNA can be exploited to construct constant-torque wind-up motors and force-torque converters.