  We reevaluate the hadronic contribution to the running of the QED fine
structure constant alpha(s) at s = M_Z^2. We use data from e+e- annihilation
and tau decays at low energy and at the qq-bar 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 delta_alpha^had = (277.8 +/- 2.6) x 10^-4 yielding
alpha^-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_mu^had = (695.1 +/- 7.5) x 10^-10.
