Using the OPAL detector at LEP, the running of the effective QED coupling α(t) is measured for space-like momentum transfer from the angular distribution of small-angle Bhabha scattering. In an almost ideal QED framework, with very favourable experimental conditions, we obtain: Δα(-6.07 GeV2) - Δα(-1.81 GeV2) = (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 agrees with current evaluations of α(t). The null hypothesis that α remains constant within the above interval of -t is excluded with a significance above 5σ. Similarly, our results are inconsistent at the level of 3 σ with the hypothesis that only leptonic loops contribute to the running. This is currently the most significant direct measurement where the running α(t) is probed differentially within the measured t range.