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The leptonic tensor for the process $e^+ e^- \to \gamma+\gamma^*$, whichdescribes the next-to-leading order virtual and soft QED corrections to initialstate radiation in $e^+ e^-$ annihilation with emission of an extra virtualphoton decaying into hadrons, is calculated. A Monte Carlo generator for thereaction $e^+ e^- \to \gamma + \pi^+ \pi^-$ has been set up which includesthese corrections. It thus describes configurations where the invariant mass ofthe hadrons plus photon is very close to $\sqrt{s}$. Predictions for cmsenergies of 1 to 10 GeV, corresponding to the energies of DAPHNE and B-mesonfactories, are presented. The possibility for an accurate measurement usingtagged photons of $\sigma(e^+ e^- \to hadrons)$, which plays an important rolein the theoretical description of the muon anomalous magnetic moment and therunning of the electromagnetic coupling, is discussed.

NLO QED corrections to ISR in $e^+ e^-$ annihilation and the measurement of $\sigma(e^+ e^- \rightarrow{\mathrm{hadrons}})$ using tagged photons