The structure of a cationic worm-like cylindrical micelle was investigated by SANS (small-angle neutron scattering). Intensities from 0.1% by weight solutions in D2O, at rest and under shear, were measured on the NIST Cold Neutron Research Facility 30-m spectrometer in the wave vector range 0.03 less than or equal to Q (nm(-1)) less than or equal to 2.0. Scattered intensity patterns from the solutions subjected to shears equal to or greater than 40 s(-1) showed pronounced anisotropy, but such anisotropy could not be detected below this apparent threshold shear. The threshold was characterized by a relaxation time since anisotropy was detected only after several minutes of shearing. In contrast, the anisotropy was apparent immediately the shear was applied at the higher shears. The data were analyzed based on the assumption that the micelles behave as rigid rods. Estimates of the radii and length under shear are given. Polydispersity in rod length is discussed, and we argue that it contributes significantly to the scattering patterns.