  We adapt the notion of the Darboux transformation to the context of
polynomial Sturm-Liouville problems. As an application, we characterize the
recently described $X_m$ Laguerre polynomials in terms of an isospectral
Darboux transformation. We also show that the shape-invariance of these new
polynomial families is a direct consequence of the permutability property of
the Darboux-Crum transformation.
