The effect of additive noise on a static front that connects a stable homogeneous state with an also stable but spatially periodic state is studied. Numerical simulations show that noise induces front propagation. The conversion of random fluctuations into direct motion of the front's core is responsible of the propagation; noise prefers to create or remove a bump, because the necessary perturbations to nucleate or destroy a bump are different. From a prototype model with noise, we deduce an adequate equation for the front's core. An analytical expression for the front velocity is deduced, which is in good agreement with numerical simulations.