The gamma-decay of the anti-analog of the giant dipole resonance (AGDR) has been measured to the isobaric analog state excited in the p(124Sn,n) reaction at a beam energy of 600 MeV/nucleon. The energy of the transition was also calculated with state-of-the-art self-consistent random-phase approximation (RPA) and turned out to be very sensitive to the neutron-skin thickness (DeltaR_(pn)). By comparing the theoretical results with the measured one, the DeltaR_(pn) value for 124Sn was deduced to be 0.175 pm 0.048 fm, which agrees well with the previous results. The energy of the AGDR measured previously for ^(208)Pb was also used to determine the DeltaR_(pn) for ^(208)Pb. In this way a very precise DeltaR_(pn) = 0.181 pm 0.031 neutron-skin thickness has been obtained for 208Pb. The present method offers new possibilities for measuring the neutron-skin thicknesses of very exotic isotopes.
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