We show that the propagation of a front into an unstable medium is qualitatively affected by the presence of noise in the case of dynamics generated by a nonconserved time-dependent Ginzburg-Landau model. In the absence of noise, well-known, mathematically rigorous results establish that fronts propagate with constant velocity. Noise, represented by nonzero temperature fluctuations, generates domain growth ahead of the front. The competition between the propagating front and bulk domains leads to a very sharp crossover to diffusive motion for the front and a velocity which decreases in time as t-1/2. These results are obtained by numerical simulations and are in agreement with a theoretical calculation using recently developed techniques.