  A high-resolution study of the electromagnetic response of $^{206}$Pb below
the neutron separation energy is performed using a ($\vec{\gamma}$,$\gamma'$)
experiment at the HI$\vec{\gamma}$S facility. Nuclear resonance fluorescence
with 100% linearly polarized photon beams is used to measure spins, parities,
branching ratios, and decay widths of excited states in $^{206}$Pb from $4.9$
to $8.1$MeV. The extracted $\Sigma$$B$(E1)$\uparrow$ and
$\Sigma$$B$(M1)$\uparrow$ values for the total electric and magnetic dipole
strength below the neutron separation energy are 0.9$\pm$0.2e$^2$fm$^2$ and
8.3$\pm$2.0$\mu_{N}^2$, respectively. These measurements are found to be in
very good agreement with the predictions from an energy-density functional
(EDF) plus quasiparticle phonon model (QPM). Such a detailed theoretical
analysis allows to separate the pygmy dipole resonance from both the tail of
the giant dipole resonance and multi-phonon excitations. Combined with earlier
photonuclear experiments above the neutron separation energy, one extracts a
value for the electric dipole polarizability of $^{206}$Pb of
$\alpha_{D}\!=\!122\pm10$mb/MeV. When compared to predictions from both the
EDF+QPM and accurately calibrated relativistic EDFs, one deduces a range for
the neutron-skin thickness of $R_{\rm skin}^{206}\!=\!0.12$-$0.19$fm and a
corresponding range for the slope of the symmetry energy of $L\!=\!48$-$60$MeV.
This newly obtained information is also used to estimate the
Maxwellian-averaged radiative cross section $^{205}$Pb(n,$\gamma$)$^{206}$Pb at
30keV to be $\sigma\!=\!130\!\pm\!25$mb. The astrophysical impact of this
measurement--on both the s-process in stellar nucleosynthesis and on the
equation of state of neutron-rich matter--is discussed.
