We present the first precision measurement of the spin-dependent asymmetry in the threshold region of (3)(H) over right arrowe((e) over right arrow ,e') at Q(2) 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 Q(2) = 0.1 (GeV/c)(2), while a small discrepancy at Q(2) = 0.2 (GeV/c)(2) is observed.