Finely sampled maps of the (C-13)O J = 1-0 emission from five dark clouds within the Taurus molecular cloud complex were made, and polarization measurements of the optical emission were obtained from background and embedded stars to determine the direction of the interstellar magnetic field towards these regions. The clouds are found to have flattened morphologies with the direction of their minor axis parallel to the direction of the magnetic field, as expected if the lateral contraction of the gas is inhibited by magnetic pressures. In addition, each cloud appears to be rotating about an axis parallel to the magnetic field direction as predicted by detailed calculations of the magnetic braking process. It is believed that the magnetic field has played a significant part in the evolution of these regions, which are characterized by mean densities of about 1000-5000/cu cm. In addition, the clouds are determined to be in virial equilibrium based on the extent and mean line-width of (C-13)O J = 1-0 emission and the masses derived from molecular hydrogen column densities.