When a quantum wire is weakly confined, a conductance plateau appears at e;{2}/h with decreasing carrier density in zero magnetic field accompanied by a gradual suppression of the 2e;{2}/h plateau. Applying an in-plane magnetic field B_{ parallel} does not alter the value of this quantization; however, the e;{2}/h plateau weakens with increasing B_{ parallel} up to 9 T, and then strengthens on further increasing B_{ parallel}, which also restores the 2e;{2}/h plateau. Our results are consistent with spin-incoherent transport in a one-dimensional wire.