  MUonE is a proposed experiment designed to measure the hadronic vacuum
polarization contribution to muon $g-2$ through elastic $\mu-e$ scattering. As
such it employs an extremely high-resolution tracking apparatus. We point out
that this makes MUonE also a very promising experiment to search for displaced
vertices from light, weakly-interacting new particles. We demonstrate its
potential by showing how it has excellent sensitivity to dark photons in the
mass range $10~\mathrm{MeV} \le m_{A'} \le 100~\mathrm{MeV}$ and kinetic mixing
parameter $10^{-5} \le \epsilon e \le 10^{-3}$, through the process
$\mu^{\pm}\, e^- \to \mu^{\pm}\, e^-\, A'$ followed by $A'\to e^+e^-$.
