We prove that among all doubly connected domains of ℝ n of the form B1\ B̅ 2, where B1 and B2 are open balls of fixed radii such that B̅ 2⊂ B1, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.