We derive the elastic theory suitable for describing the free energy required to deform a stiff helical molecule with the symmetry of DNA. At quadratic levels in the strains we find, in addition to the independent bending and torsional energies incorporated in previous theories, a previously unknown coupling between twist and bend. If the backbone is given constant curvature, minimization of the free energy with respect to the twist degrees of freedom indicates that this coupling drives a decrease in the molecular twist, or an unwinding of the helix. New experiments are proposed to bring out the symmetry-breaking effecTs of the twist-bend coupling : (i) ring closure experiments will indicate a helix repeat that becomes progressively more underwound for smaller rings, and (ii) gel mobilities of supercoiled rings of integral-helix-repeat length, with equal and opposite added linking numbers, will differ.