A stressed fiber-optic waveguide is used to impart orbital angular momentum to a Hermite-Gaussian (HG(10)) laser mode. The transmitted beam has an annular intensity profile and a well-defined azimuthal phase dependence. We confirm the phase structure of the beam by observing the interference pattern produced between it and a plane wave. The transfer of the angular momentum to the light occurs because of a difference in phase velocity within the fiber for two orthogonal modes that comprise the input beam. This represents a mechanism for the transfer of orbital angular momentum to a light beam that has not hitherto been identified.