  We obtain eigenvalues and eigenfunctions of the Schr\"{o}dinger equation with
a hyperbolic double-well potential. We consider exact polynomial solutions for
some particular values of the potential-strength parameter and also numerical
energies for arbitrary values of this model parameter. We test the numerical
method by means of a suitable exact asymptotic expression for the eigenvalues
and also calculate critical values of the strength parameter that are related
to the number of bound states supported by the potential.
