Herrn Professor Helmut Grunsky gewidmet In an earlier joint paper of the first author, the following theorem was “Supposefis a nonconstant meromorphic function in the complex plane, that shares two finite nonzero values counting multiplicities with one of its derivatives. Thenfis identical with that derivative.” In this paper we settle the case that one of the shared values is allowed to be zero. The claim is then the same.