We consider the propagation of slow light with an orbital angular momentum (OAM) in a moving atomic medium. We have derived a general equation of motion and applied it in analyzing propagation of slow light with an OAM in a rotating medium, such as a vortex lattice. We have shown that the OAM of slow light manifests itself in a rotation of the polarization plane of linearly polarized light. To extract a pure rotational phase shift, we suggest to measure a difference in the angle of the polarization plane rotation by two consecutive light beams with opposite OAM. The differential angle ? al is proportional to the rotation frequency of the medium ?rot and the winding number l of light, and is inversely proportional to the group velocity of light. For slow light the angle ? al should be large enough to be detectable. The effect can be used as a tool for measuring the rotation frequency ?rot of the medium. © 2007 The American Physical Society.