In a recent paper, Sternberg and Knowles characterized implicitly the solution of a relaxed Saint-Venant flexure problem that is associated with the absolute minimum of the total strain energy among all solutions of this relaxed problem that correspond to a fixed resultant load and to the normal tractions on the ends of the cylinder inherent in Saint-Venant’s solution. In the present investigation, this optimal flexure solution is determined explicitly for a circular cylinder by means of the Papkovich-Neuber stress functions. The results obtained, which are in infinite-series form, are evaluated numerically and compared with the analogous results of Saint-Venant. The solution deduced here also supplies a quantitative illustration of Saint-Venant’s principle.