In this article we examine tile problems of boundary control and stabilization for a one-dimensional wave equation with interior point masses. We show that singularities in waves are "smoothed one order" as they cross a point mass. Thus in the case of one interior point mass, with, e.g., L(2)-Dirichlet control at the left end, the most general reachable space (from 0) that one can expect is L(2) x H-1 to the left of the mass and H-1 x L(2) to the right of the mass. We show that this is in fact the optimal result (module certain compatibility conditions). Several related results for both control and stabilization of such systems are also given.