  We study the $Z\rightarrow \gamma \bar \gamma$ process in which the $Z$ boson
decays into a photon $\gamma$ and a massless dark photon $\bar \gamma$, when
the latter couples to standard-model fermions via dipole moments. This is a
simple yet nontrivial example of how the Landau-Yang theorem---ruling out the
decay of a massive spin-1 particle into two photons---is evaded if the final
particles can be distinguished. The striking signature of this process is a
resonant monochromatic single photon in the $Z$-boson center of mass together
with missing momentum. LEP experimental bounds allow a branching ratio up to
about 10$^{-6}$ for such a decay. In a simplified model of the dark sector, the
dark-photon dipole moments arise from one-loop exchange of heavy dark fermions
and scalar messengers. The corresponding prediction for the rare $Z\to \gamma
\bar \gamma$ decay width can be explored with the large samples of $Z$ bosons
foreseen at future colliders.
