The second-order perturbative treatment of molecular electronic energies is modified in a way that takes account of near-degeneracy effects. The resulting formalism (DCPT2) is just as easy to apply as the popular second-order Moller-Plesset theory (also called second-order many-body perturbation theory) but is more accurate because it is less sensitive to near-degeneracy effects. Illustrative calculations are reported for a series of small molecules, showing significant improvement for cases with lone pairs on vicinal atoms.