  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.
