  The $^{208}$Pb($p$,$n\gamma\bar p$) $^{207}$Pb reaction at a beam energy of
30 MeV has been used to excite the anti-analog of the giant dipole resonance
(AGDR) and to measure its $\gamma$-decay to the isobaric analog state in
coincidence with proton decay of IAS. The energy of the transition has also
been calculated with the self-consistent relativistic random-phase
approximation (RRPA), and found to be linearly correlated to the predicted
value of the neutron-skin thickness ($\Delta R_{pn}$). By comparing the
theoretical results with the measured transition energy, the value of 0.190
$\pm$ 0.028 fm has been determined for $\Delta R_{pn}$ of $^{208}$Pb, in
agreement with previous experimental results. The AGDR excitation energy has
also been used to calculate the symmetry energy at saturation ($J=32.7 \pm 0.6$
MeV) and the slope of the symmetry energy ($L=49.7 \pm 4.4$ MeV), resulting in
more stringent constraints than most of the previous studies.
