  We consider a maximization problem for eigenvalues of the Laplace-Beltrami
operator on surfaces of revolution in $\mathbb{R}^3$ with two prescribed
boundary components. For every $j$, we show that there is a surface $\Sigma_j$
which maximizes the $j$-th Dirichlet eigenvalue. The maximizing surface has a
meridian which is a rectifiable curve. If there is a catenoid which is the
unique area minimizing surface with the prescribed boundary, then the
eigenvalue maximizing surfaces of revolution converge to this catenoid.
