  We calculate the effective mass of the $\omega$ meson in nuclear matter in a
relativistic random-phase approximation to the Walecka model. The dressing of
the meson propagator is driven by its coupling to particle-hole pairs and
nucleon-antinucleon ($N\bar{N}$) excitations. We report a reduction in the
$\omega$-meson mass of about 170~MeV at nuclear-matter saturation density. This
reduction arises from a competition between the density-dependent
(particle-hole) dressing of the propagator and vacuum polarization ($N\bar{N}$
pairs). While density-dependent effects lead to an increase in the mass
proportional to the classical plasma frequency, vacuum polarization leads to an
even larger reduction caused by the reduced effective nucleon mass in the
medium.
