The theory of quantum electrodynamics (QED) predicts that beta decay of the neutron into a proton, electron and antineutrino should be accompanied by a continuous spectrum of soft photons. While this inner bremsstrahlung branch has been previously measured in nuclear beta and electron capture decay, it has never been observed in free neutron decay. Recently, the photon energy spectrum and branching ratio for neutron radiative decay have been calculated using two approaches: a standard QED framework and heavy baryon chiral perturbation theory (an effective theory of hadrons based on the symmetries of quantum chromodynamics). The QED calculation treats the nucleons as point-like, whereas the latter approach includes the effect of nucleon structure in a systematic way. Here we observe the radiative decay mode of free neutrons, measuring photons in coincidence with both the emitted electron and proton. We determined a branching ratio of (3.13 +/- 0.34) x 10(-3) (68 per cent level of confidence) in the energy region between 15 and 340 keV, where the uncertainty is dominated by systematic effects. The value is consistent with the predictions of both theoretical approaches; the characteristic energy spectrum of the radiated photons, which differs from the uncorrelated background spectrum, is also consistent with the calculated spectrum. This result may provide opportunities for more detailed investigations of the weak interaction processes involved in neutron beta decay.