  We have measured the running of the effective QED coupling constant
$\alpha(s)$ in the time-like region $0.6<\sqrt s< 0.975$ GeV with the KLOE
detector at DA$\Phi$NE using the Initial State Radiation process
$e^+e^-\to\mu^+ \mu^-\gamma$. It represents the first measurement of the
running of $\alpha(s)$ in this energy region. Our results show a more than
5$\sigma$ significance of the hadronic contribution to the running of
$\alpha(s)$, which is the strongest direct evidence both in time- and
space-like regions achieved in a single measurement. By using the
$e^+e^-\to\pi^+\pi^-$ cross section measured by KLOE, the real and imaginary
part of the shift $\Delta\alpha(s)$ has been extracted. By a fit of the real
part of $\Delta\alpha(s)$ and assuming the lepton universality the branching
ratio $BR(\omega\to\mu^+\mu^-) = (6.6\pm1.4_{stat}\pm1.7_{syst})\cdot 10^{-5} $
has been determined.
