In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. %either a totally geodesic disk or a spherical cap.
Our result also provides a proof
of a conjecture proposed by Sternberg-Zumbrun in {\it J Reine Angew Math 503 (1998), 63--85}.
We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary
in a ball, which, together with the new Minkowski formula,
yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.