The shear-alignment process of the hexagonal lyotropic liquid-crystalline phase formed by a solution of hexa(ethylene glycol) monododecyl ether (C(12)E(6)) in heavy water is examined. Experiments were performed with a novel cone-and-plate rheo-NMR apparatus, which allows the simultaneous measurement of shear viscosity and deuterium nuclear magnetic resonance. The NMR spectra show that initially disordered samples having polydomain structures become macroscopically aligned during shear. The aligned state shows no relaxation after cessation of shear. With proceeding shear alignment, the apparent shear viscosity decreases. The changes of the deuterium NMR spectra observed during the alignment process reveal that the reorientation of the domains depends not on the shear rate but on the shear strain. NMR experiments using different shear geometries show that at shear rates of ca. 2 s(-1) the hexadic axis becomes aligned parallel to the flow. At lower shear rates, we cannot yet distinguish between an orientation of he micelles in the flow direction or perpendicular to the shear plane; an orientation along the now direction, however, appears more likely.