Using the OPAL detector at LEP, the running of the effective QED coupling alpha(t) is measured for space-like momentum transfer through its effect on the angular spectrum of small-angle Bhabha scattering. In an almost ideal QED framework, with very favourable experimental conditions, we obtain: delta alpha(-6.07 GeV^2) - delta alpha(-1.81 GeV^2) = (440 +- 58 +- 43 +- 30) * 10^{-5} where the first error is statistical, the second is the experimental systematic and the third is the theoretical uncertainty. This is the strongest direct evidence ever presented that the running of $\alpha$ is consistent with Standard Model expectations. The null hypothesis that alpha remains constant within the above interval of -t is excluded with a significance above 5 sigma. Similarly, our results are inconsistent at the level of 3 sigma with the hypothesis that only leptonic loops contribute to the running, and therefore provide the first clear experimental evidence that hadronic loops also contribute.