  A search for the rare decay $D^0 \to \mu^+ \mu^-$is performed using a data
sample, corresponding to an integrated luminosity of 0.9 fb$^{-1}$, of $pp$
collisions collected at a centre-of-mass energy of 7 TeV by the LHCb
experiment. The observed number of events is consistent with the background
expectations and corresponds to an upper limit of \mbox{${\cal B}(D^0 \to \mu^+
\mu^-) < 6.2(7.6) \times 10^{-9}$} at \mbox{90%} \mbox{(95%)} confidence level.
This result represents an improvement of more than a factor twenty with respect
to previous measurements.
