A measurement of the double-spin asymmetry from electron-^3He scattering in the threshold region of two- and three-body breakup of ^3He was performed at Jefferson Lab, for Q^2 values of 0.1 and 0.2 (GeV/c)^2. The results of this measurement serve as a stringent test of our understanding of few-body systems. When compared with calculations from plane wave impulse approximation and Faddeev theory, we found that the Faddeev calculations, which use modern nuclear potentials and prescriptions for meson-exchange currents, demonstrate an overall good agreement with data.