  A search for the lepton flavour violating decay $\tau^-\to \mu^-\mu^+\mu^-$
is performed with the LHCb experiment. The data sample corresponds to an
integrated luminosity of $1.0\mathrm{\,fb}^{-1}$ of proton-proton collisions at
a centre-of-mass energy of $7\mathrm{\,Te\kern -0.1em V}$ and
$2.0\mathrm{\,fb}^{-1}$ at $8\mathrm{\,Te\kern -0.1em V}$. No evidence is found
for a signal, and a limit is set at $90\%$ confidence level on the branching
fraction, $\mathcal{B}(\tau^-\to \mu^-\mu^+\mu^-) < 4.6 \times 10^{-8}$.
