The COMPASS collaboration at CERN has investigated pion Compton scattering, π-γ→π-γ, at center-of-mass energy below 3.5 pion masses. The process is embedded in the reaction π-Ni→π-γNi, which is initiated by 190 GeV pions impinging on a nickel target. The exchange of quasireal photons is selected by isolating the sharp Coulomb peak observed at smallest momentum transfers, Q2<0.0015  (GeV/c)2. From a sample of 63 000 events, the pion electric polarizability is determined to be απ=(2.0±0.6stat±0.7syst)×10-4  fm3 under the assumption απ=-βπ, which relates the electric and magnetic dipole polarizabilities. It is the most precise measurement of this fundamental low-energy parameter of strong interaction that has been addressed since long by various methods with conflicting outcomes. While this result is in tension with previous dedicated measurements, it is found in agreement with the expectation from chiral perturbation theory. An additional measurement replacing pions by muons, for which the cross-section behavior is unambiguously known, was performed for an independent estimate of the systematic uncertainty.