We reevaluate the hadronic contribution to the running of the QED fine structure constant α( s) at s= M Z 2. We use data from e + e − annihilation and τ decays at low energy and at the q q ̄ thresholds, where resonances occur. Using so-called spectral moments and the Operator Product Expansion (OPE), it is shown that a reliable theoretical prediction of the hadronic production rate R( s) is available at relatively low energies. Its application improves significantly the precision on the hadronic vacuum polarization contribution. We obtain Δα had( M Z 2)=(277.8±2.6)×10 −4 yielding α −1( M Z 2)=128.923±0.036. Inserting this value in a global electroweak fit using current experimental input, we constrain the mass of the Standard Model Higgs boson to be M Higgs =129 +103 −62 GeV . Analogously, we improve the precision of the hadronic contribution to the anomalous magnetic moment of the muon for which we obtain a μ had=(695.1±7.5)×10 −10.