  A search for the very rare $D^0 \to \mu^+\mu^-$ decay is performed using data
collected by the LHCb experiment in proton-proton collisions at $\sqrt{s} = 7$,
8 and $13~\rm{TeV}$, corresponding to an integrated luminosity of
$9~\rm{fb^{-1}}$. The search is optimised for $D^0$ mesons from $D^{\ast+}\to
D^0\pi^+$ decays but is also sensitive to $D^0$ mesons from other sources. No
evidence for an excess of events over the expected background is observed. An
upper limit on the branching fraction of this decay is set at $\mathcal{B}(D^0
\to \mu^+\mu^-) < 3.1 \times 10^{-9}$ at a $90\%$ CL. This represents the
world's most stringent limit, constraining models of physics beyond the
Standard Model.
