  The evolution of the electromagnetic coupling, alpha, in the
momentum-transfer range 1800GeV^2 < -Q^2 < 21600GeV^2 is studied with about
40000 Bhabha-scattering events collected with the L3 detector at LEP at
centre-of-mass energies 189-209GeV. The running of alpha is parametrised as:
alpha(Q^2) = alpha_0/(1-C Delta alpha(Q^2)), where alpha_0=\alpha(Q^2=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 theta|<0.9 excludes the hypothesis of a constant value
of alpha, 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.
