We present the first precision measurement of the spin-dependent asymmetry in the threshold region of 3He(e,e') at Q2 values of 0.1 and 0.2 (GeV/c)2. The agreement between the data and nonrelativistic Faddeev calculations which include both final-state interactions and meson-exchange current effects is very good at Q2 = 0.1 (GeV/c)2, while a small discrepancy at Q2 = 0.2 (GeV/c)2 is observed.