We have evaluated the bulk and surface elastic constants of a nematic liquid crystal composed of Gay-Berne particles, using a generalization of earlier theories by Poniewierski and Stecki, and Lipkin, Rice and Mohanty. The ratios between surface and bulk elastic constants have been plotted as a function of temperature and at nematic-isotropic coexistence. In particular, the sign of the K ₁₃ (splay-bend) constant has been shown to depend sensitively on details of the intermolecular potential. Results have been contrasted with those obtained from a different theory.