We consider the directed Abelian sandpile model in the presence of sink sites whose density f_t at depth t below the top surface varies as ct^(-x). For x>1 the disorder is irrelevant. For x<1, it is relevant and the model is no longer critical for any nonzero c. For x=1 the exponents of the avalanche distributions depend continuously on the amplitude c of the disorder. We calculate this dependence exactly, and verify the results with simulations.