We report a measurement of the asymmetry in spin-dependent quasielastic scattering of longitudinally polarized electrons from a polarized $^{3}\mathrm{He}$ gas target. The asymmetry is measured at kinematics sensitive to the transverse-longitudinal response function ${\mathit{R}}_{\mathit{T}\mathit{L}}$(${\mathit{Q}}^{2}$,\ensuremath{\omega}). The value of the neutron electric form factor ${\mathit{G}}_{\mathit{E}}^{\mathit{n}}$(${\mathit{Q}}^{2}$=0.16 (GeV/${\mathit{c}}^{2}$))=+0.070\ifmmode\pm\else\textpm\fi{}0.100\ifmmode\pm\else\textpm\fi{}0.035 is extracted from the asymmetry using a Faddeev calculation of the $^{3}\mathrm{He}$ wave function.