  The leptonic tensor for the process $e^+ e^- \to \gamma+\gamma^*$, 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 $e^+ e^- \to \gamma + \pi^+ \pi^-$ 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 $\sigma(e^+ e^- \to hadrons)$, which plays an important role
in the theoretical description of the muon anomalous magnetic moment and the
running of the electromagnetic coupling, is discussed.

High Energy Physics - Phenomenology