We report on the first dispersive calculation of the pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon $(g-2)_\mu$. It is unambiguously defined in terms of the doubly-virtual pion transition form factor, which is in principle directly accessible in experiment. In the absence of a complete measurement covering all kinematic regions relevant for $(g-2)_\mu$, we derive a dispersive representation that accounts for all the low-lying singularities, reproduces the correct high- and low-energy limits, and proves suitable for the evaluation of the $(g-2)_\mu$ loop integral based on the available data for $\pi^0\to\gamma\gamma$, $e^+e^-\to 3\pi$, and $e^+e^-\to e^+e^-\pi^0$. Our final result as a complete data-driven determination with fully controlled uncertainty estimates, $a_\mu^{\pi^0\text{-pole}}=62.6^{+3.0}_{-2.5}\times 10^{-11}$, can be further improved in light of forthcoming singly-virtual measurements.