We have studied the one-meson radiative tau decays À ! À ðK À Þ, computing the structuredependent contributions within a Lagrangian approach based on the large-N C limit of QCD that ensures the proper low-energy limit dictated by chiral symmetry. Upon imposing the short-distance QCD constraints to the form factors, we are able to predict the structure-dependent radiation without any free parameter and, therefore, the relevant observables for the decay À ! À : the photon energy spectrum, the invariant mass spectrum of the meson-photon system, and the integrated decay rate. We also discuss the remaining uncertainties in these observables for the À ! K À decay. According to our results, the present facilities could detect these rare decays for the first time in the near future, allowing for a stringent test of our predictions.