Menger's theorem for infinite graphs

We prove that Menger's theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set P of disjoint A-B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in P. This settles an old conjecture of Erdos.