We study a boundary perturbation problem for a one-dimensional Schrödinger equation in which the potential has a regular singularity near the perturbed end point. We give the asymptotic behaviour of the eigenvalues under the perturbation. This problem arose out of the author's studies of singular elliptic operators in higher dimensions and we illustrate this point with an example. The class of potentials to which this method applies is larger than that covered by standard results, which assume uniform ellipticity of the operator or a perturbative term that is analytic in the perturbation parameter.