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A method for determining the radii of excited states of nuclei by means of ([sup.3]He, t) chargeexchange reactions was proposed. Two versions of a comparison of differential cross sections for ([sup.3]He, t) reactions were considered. The first relies on a comparison with cross sections for inelastic-scattering processes leading to the formation of isobaric analog states, while the second involves ([sup.3]He, t) reactions leading to the production of the ground state. The two versions in question yield similar results and make it possible to determine the radius of the first excited state of the [sup.13]N nucleus. This state has the excitation energy of [E.sup.*] = 2.37 MeV, lying above the proton-emission threshold. The resulting radius proved to be enhanced in relation to the ground state and is close to the radius of the 3.09 -MeV isobaric analog state of the [sup.13]C nucleus, which has a neutron halo. This permitted drawing the conclusion that the [sup.13]N nucleus in the 2.37-MeV state has a proton halo. The possibility of revealing a proton halo in other states of light nuclei is considered.