  I calculate the QED coupling, alpha, directly in the MS-bar scheme using an
unsubtracted dispersion relation for the three light quarks, and perturbative
QCD for charm and bottom quarks. Compact analytical expressions are presented,
making this approach particularly suitable for electroweak fits. After
alpha^(-1) (m_tau) = 133.513 +- 0.025 is obtained in a first step, I perform a
4-loop renormalization group evolution with 3-loop matching conditions to
arrive at alpha^(-1) (M_Z) = 127.934 +- 0.026 for alpha_s (M_Z) = 0.120. The
corresponding hadronic contribution to the on-shell coupling is Delta
alpha_had^(5) (M_Z) = 0.02779 +- 0.00019. The error is mainly from m_c, and
from experimental uncertainties in e^+ e^- annihilation into unflavored and
strange hadrons and tau decay data.
