We report a measurement of the asymmetry in spin-dependent quasielastic scattering of longitudinally polarized electrons from a polarized 3He gas target. This measurement represents the first demonstration of a new method for studying electromagnetic nuclear structure: the scattering of polarized electrons from a polarized nuclear target. The measured asymmetry is in good agreement with a Faddeev calculation and supports the picture of spin-dependent quasielastic scattering from polarized 3He as predominantly scattering from a polarized neutron.