  We prove a uniqueness theorem for immersed spheres of prescribed
(non-constant) mean curvature in homogeneous three-manifolds. In particular,
this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed
spheres of prescribed Weingarten curvature in R3 for the special but important
case of prescribed mean curvature. As a consequence, we extend the classical
Hopf uniqueness theorem for constant mean curvature spheres to the case of
immersed spheres of prescribed antipodally symmetric mean curvature in R3.
