This Letter reports the observation of $\tau$-lepton-pair production in ultraperipheral lead-lead collisions $\mathrm{Pb}+\mathrm{Pb}\to \mathrm{Pb}(\gamma \gamma \to \tau \tau )\mathrm{Pb}$ and constraints on the $\tau$-lepton anomalous magnetic moment $a_{\tau }$. The dataset corresponds to an integrated luminosity of $1.44{\mathrm{nb}}^{-1}$ of LHC $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s_{\mathrm{NN}}}=5.02\mathrm{TeV}$ recorded by the ATLAS experiment in 2018. Selected events contain one muon from a $\tau$-lepton decay, an electron or charged-particle track(s) from the other $\tau$-lepton decay, little additional central-detector activity, and no forward neutrons. The $\gamma \gamma \to \tau \tau$ process is observed in $\mathrm{Pb}+\mathrm{Pb}$ collisions with a significance exceeding 5 standard deviations and a signal strength of ${\mu }_{\tau \tau }=1.03_{-0.05}^{+0.06}$ assuming the standard model value for $a_{\tau }$. To measure $a_{\tau }$, a template fit to the muon transverse-momentum distribution from $\tau$-lepton candidates is performed, using a dimuon ($\gamma \gamma \to \mu \mu$) control sample to constrain systematic uncertainties. The observed 95% confidence-level interval for $a_{\tau }$ is $-0.057<a_{\tau }<0.024$.