  We compute the class groups of very general normal surfaces in complex
projective three-space containing an arbitrary base locus $Z$, thereby
extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty).
Our method is an adaptation of Griffiths and Harris' degeneration proof,
simplified by a cohomology and base change argument. We give applications to
computing Picard groups, which generalize several known results.
