In previous work we introduced the concept of black-hole entropy, which we identified with the surface area of the black hole in question expressed in units of the Planck length squared. We suggested that the appropriate generalization of the second law for a region containing a black hole is that the black-hole entropy plus the common entropy in the black-hole exterior never decreases. Here we establish the validity of this law for the infall of an entropy-bearing system into a much larger and more massive generic stationary black hole. To do this we determine a general lower bound for the increase in black-hole entropy, and an upper bound for the entropy of the system, while allowing for quantum effects at each stage. In passing we show that the generalized second law is a statistical law which becomes over-whelmingly probable in the limit of a macroscopic system. We also consider briefly more general situations. Finally, we give two simple examples of predictions made by the generalized second law for black-hole formation processes.