Starting from the stochastic Landau-Lifschitz-Gilbert equation, we derive Langevin equations that describe the nonzero-temperature dynamics of a rigid domain wall. We derive an expression for the average drift velocity of the domain wall r(dw) as a function of the applied current, and find qualitative agreement with recent magnetic semiconductor experiments. Our model implies that at any nonzero-temperature r(dw) initially varies linearly with current, even in the absence of nonadiabatic spin torques.