In [\textit{F. A. Grünbaum} and \textit{L. Haine}, Symmetries and Integrability of Differential Equations, Estérel, 1994, CRM Proc. Lect. Notes 9, 143-154 (1996; Zbl 0865.33008)] the Darboux transform was used to obtain so-called Bochner-Krall orthogonal polynomials which satisfy a higher order (\(>2\)) spectral type differential equation. This Darboux transform is based on the factorization of the second order difference operator induced by the three term recurrence relation into a product of two first order difference operators and was explicitly applied to the classical orthogonal polynomials by Grünbaum and Haine to produce examples of Bochner-Krall orthogonal polynomials. In the paper under review this Darboux process is studied in a more general setting. Conditions are obtained for the parameters involved under which this Darboux transform will yield another orthogonal polynomial system. In a sense the main result explains that the ``Krall polynomials'' obtained by Grünbaum and Haine can be obtained from the classical orthogonal polynomials using suitable Darboux transforms since the ``Krall polynomials'' are orthogonal with respect to a classical moment functional plus one or two point masses.