  The precision measurement of the $d\sigma(e^+e^- \rightarrow \mu^+\mu^-
\gamma)/d\sqrt{s}$ cross section with the photon emitted in the initial state
with the KLOE detector has been used to measure the running of the QED coupling
constant $\alpha(s)$ in the energy range 0.6 $< \sqrt{s} <$ 0.975 GeV in the
time-like region. We were able to achieve a significance of the hadronic
contribution to the running of $\alpha(s)$ of more than 5$\sigma$ with a clear
contribution of the $\rho - \omega$ resonances to the photon propagator. The
real and imaginary part of the shift $\Delta\alpha$ to the running has been
estracted and a fit of the real part allowed us to measure the branching
fraction $BR(\omega \rightarrow \mu^+\mu^-)$ =
(6.6$\pm$1.4$_{stat}$$\pm$1.7$_{syst}$ )$\cdot$10$^{-5}$.
