The leptonic tensor for the process $e^+ e^- \rightarrow \gamma+\gamma^*$,
which describes the next-to-leading order virtual and soft QED corrections to
initial state radiation in $e^+ e^-$ annihilation with emission of an extra
virtual photon decaying into hadrons, is calculated. A Monte Carlo generator
for the reaction $e^+ e^- \rightarrow \gamma + \pi^+ \pi^-$ has been set up
which includes these corrections. It thus describes configurations where the
for the reaction $e^+ e^- \rightarrow \gamma + \pi^+ \pi^-$ has been set up
which includes these corrections. It thus describes configurations where the
invariant mass of the hadrons plus photon is very close to $\sqrt{s}$.
Predictions for cms energies of 1 to 10 GeV, corresponding to the energies of
DAPHNE and B-meson factories, are presented. The possibility for an accurate
measurement using tagged photons of $\sigma(e^+ e^- \rightarrow hadrons)$,
which plays an important role in the theoretical description of the muon
anomalous magnetic moment and the running of the electromagnetic coupling, is
discussed.