  We present an observation of photon-photon production of $\tau$ lepton pairs
in ultraperipheral lead-lead collisions. The measurement is based on a data
sample with an integrated luminosity of 404 $\mu$b$^{-1}$ collected by the CMS
experiment at a nucleon-nucleon center-of-mass energy of 5.02 TeV. The
$\gamma\gamma$ $\to$ $\tau^+\tau^-$ process is observed for $\tau\tau$ events
with a muon and three charged hadrons in the final state. The measured fiducial
cross section is $\sigma(\gamma\gamma$ $\to$ $\tau^+\tau^-)$ = 4.8 $\pm$ 0.6
(stat) $\pm$ 0.5 (syst) $\mu$b, in agreement with leading-order QED
predictions. Using $\sigma(\gamma\gamma$ $\to$ $\tau^+\tau^-)$, we estimate a
model-dependent value of the anomalous magnetic moment of the $\tau$ lepton of
$a_\tau$ = 0.001 $^{+0.055}_{-0.089}$.
