The evolution of the electromagnetic coupling, α, in the momentum-transfer range 1800 GeV2 < -Q2 < 21600 GeV2 is studied with about 40 000 Bhabha-scattering events collected with the L3 detector at LEP at centre-of-mass energies √s = 189-209 GeV. The running of α is parametrised as: α(Q2) = α0/1 - C Δα(Q2), where α ≡ α(Q2 = 0) is the fine-structure constant and C = 1 corresponds to the evolution expected in QED. A fit to the differential cross section of the e+e- → e+e- process for scattering angles in the range |cos θ| < 0.9 excludes the hypothesis of a constant value of α, C = 0, and validates the QED prediction with the result: C = 1.05±0.07±0.14, where the first uncertainty is statistical and the second systematic. © 2005 Elsevier B.V. All rights reserved.