We introduce and analyze the longitudinal-polarization asymmetry of the process ${e}^{+}$${e}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}hadrons with longitudinally polarized electron beams on and near the ${Z}^{0}$ resonance. We show that, in spite of the intrinsic strong-interaction presence in the final state, the vast majority of the diagrams which contribute to one electroweak loop are free of strong-interaction effects. Further, on ${Z}^{0}$ resonance the asymmetry is independent of the final states, giving a commensurate increase in statistics. We show that for these reasons the total theoretical strong-interaction uncertainty on ${Z}^{0}$ resonance is \ensuremath{\Delta}${A}_{\mathrm{LR}}^{{e}^{+}{e}^{\mathrm{\ensuremath{-}}}\ensuremath{\rightarrow}\mathrm{hadrons}\ensuremath{\lesssim}0.004}$, allowing a measurement of ${A}_{\mathrm{LR}}$ to \ifmmode\pm\else\textpm\fi{}0.015 (which can be interpreted as a measurement of ${\mathrm{sin}}^{2}$${\mathrm{\ensuremath{\theta}}}_{\mathrm{W}}$ to \ifmmode\pm\else\textpm\fi{}0.002) with only \ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}${10}^{4}$ ${Z}^{0}$'s. Further, a measurement of ${A}_{\mathrm{LR}}$ to \ifmmode\pm\else\textpm\fi{}0.008, or ${\mathrm{sin}}^{2}$${\mathrm{\ensuremath{\theta}}}_{\mathrm{W}}$ to \ifmmode\pm\else\textpm\fi{}0.001, can be done with \ensuremath{\sim}${10}^{6}$ ${Z}^{0}$'s. This rather peculiar property of the asymmetry could allow Stanford Linear Collider and CERN LEP experiments to test the standard Glashow-Salam-Weinberg (GSW) theory and possible new physics beyond GSW early in the lifetime of these accelerators.