  The first observation of $\tau$ lepton pair production in ultraperipheral
nucleus-nucleus collisions, a pure quantum electrodynamics (QED) process, is
presented. The measurement is based on a data sample collected by the CMS
experiment at a per nucleon center-of-mass energy of $5.02~\mathrm{TeV}$, and
corresponding to an integrated luminosity of $404~\mu\mathrm{b}^{-1}$. The
$\gamma\gamma\to\tau^{+}\tau^{-}$ production is observed with a statistical
significance of at least five standard deviations for $\tau^{+}\tau^{-}$ events
with a muon and three charged hadrons in the final state. The cross section is
measured in a fiducial phase space region, and is found to be
$\sigma(\gamma\gamma\to\tau^{+}\tau^{-}) = 4.8\pm 0.6\,(\mathrm{stat})\pm
0.5\,(\mathrm{syst})~\mu\mathrm{b}$, in agreement with leading-order QED
predictions. The measurement, based on a small fraction of the expected
integrated luminosity of the LHC nuclear program, establishes the potential for
a substantially more precise determination of the anomalous magnetic moment of
the $\tau$ lepton, which is currently poorly constrained.
