We study the dynamics of a driven interface in a two-dimensional random-field Ising model close to the depinning transition at small but finite temperatures T using Glauber dynamics. A square lattice is considered with an interface initially in the (11)-direction. The drift velocity v is analyzed using finite-size scaling at T = 0 and additionally finite-temperature scaling close to the depinning transition. In both cases a perfect data collapse is obtained from which we deduce β ≈ 1/3 for the exponent which determines the dependence of v on the driving field, ν ≈ 1 for the exponent of the correlation length and δ ≈ 5 for the exponent which determines the dependence of v on T.