We report a precision measurement of the parity-violating asymmetry $A_{PV}$ in the elastic scattering of longitudinally polarized electrons from $^{208}$Pb. We measure $A_{PV}=550pm 16 {rm (stat)}pm 8 {rm (syst)}$ parts per billion, leading to an extraction of the neutral weak form factor $F_W(Q^2 = 0.00616 {rm GeV}^2) = 0.368 pm 0.013$. Combined with our previous measurement, the extracted neutron skin thickness is $R_n-R_p=0.283 pm 0.071$~fm. The result also yields the first significant direct measurement of the interior weak density of $^{208}$Pb: $rho^0_W = -0.0796pm0.0036 {rm (exp.)}pm0.0013 {rm (theo.)} {rm fm}^{-3}$ leading to the interior baryon density $rho^0_b = 0.1480pm0.0036 {rm (exp.)}pm0.0013 {rm (theo.)} {rm fm}^{-3}$. The measurement accurately constrains the density dependence of the symmetry energy of nuclear matter near saturation density, with implications for the size and composition of neutron stars.
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