  The second eigenvalue of the Robin Laplacian is shown to be maximal for the
ball among domains of fixed volume, for negative values of the Robin parameter
$\alpha$ in the regime connecting the first nontrivial Neumann and Steklov
eigenvalues, and even somewhat beyond the Steklov regime. The result is close
to optimal, since the ball is not maximal when $\alpha$ is sufficiently large
negative, and the problem admits no maximiser when $\alpha$ is positive.
