Electron-positron annihilation into hadrons plus an energetic photon from
initial state radiation allows the hadronic cross-section to be measured
over a wide range of energies. The full next-to-leading order QED corrections
for the cross-section for e^+ e^- annihilation into a real tagged photon and
a virtual photon converting into hadrons are calculated where the tagged
photon is radiated off the initial electron or
positron. This includes virtual and soft photon
corrections to the process e^+ e^- \to \gamma +\gamma^* and the emission
of two real hard photons: e^+ e^- \to \gamma + \gamma + \gamma^*.
A Monte Carlo generator has been constructed, which incorporates these
corrections and simulates the production of two charged pions or muons
plus one or two photons. Predictions are presented for centre-of-mass
energies between 1 and 10 GeV, corresponding to the energies of DAPHNE,
CLEO-C and B-meson factories.