We present an observation of photon-photon production of 𝜏 lepton pairs in ultraperipheral lead-lead collisions. The measurement is based on a data sample with an integrated luminosity of 404  μ⁢b−1 collected by the CMS experiment at a center-of-mass energy per nucleon pair of √𝑠𝑁𝑁=5.02  TeV. The 𝛾⁢𝛾→𝜏+⁢𝜏− process is observed for 𝜏+⁢𝜏− events with a muon and three charged hadrons in the final state. The measured fiducial cross section is 𝜎⁡(𝛾⁢𝛾→𝜏+⁢𝜏−)=4.8±0.6⁢(stat)±0.5⁢(syst)  μ⁢b, where the second (third) term corresponds to the statistical (systematic) uncertainty in 𝜎⁡(𝛾⁢𝛾→𝜏+⁢𝜏−) in agreement with leading-order QED predictions. Using 𝜎⁡(𝛾⁢𝛾→𝜏+⁢𝜏−), we estimate a model-dependent value of the anomalous magnetic moment of the 𝜏 lepton of 𝑎𝜏=0.00⁢1+0.055
−0.089.