  We calculate the rate of transverse relaxation arising from vortex motion in
the mixed state of YBa_2Cu_3O_7 with the static field applied along the c axis.
The vortex dynamics are described by an overdamped Langevin equation with a
harmonic elastic free energy. We find that the variation of the relaxation with
temperature, average magnetic field, and local field is consistent with
experiments; however, the calculated time dependence is different from what has
been measured and the value of the rates calculated is roughly two orders of
magnitude slower than what is observed. Combined with the strong experimental
evidence pointing to vortex motion as the dominant mechanism for T_2
relaxation, these results call into question a prior conclusion that vortex
motion is not significant in T_1 measurements in the vortex state.
