The classical and quantum mechanics of an electron moving in the field of an electric dipole is studied in some detail. It is shown that there exists a family of circular orbits lying on the surface of a half-cone stretching from the site of the dipole to infinity. All these orbits have the same energy and angular momentum. Although the dipole potential is highly non-central, the Schrödinger equation turns out to be separable in spherical polar coordinates. It is shown that the cone of the classical 'bound' states also develops from the angular part of the Schrödinger equation on imposing the Wilson-Sommerfeld quantization rule. Instead of a discrete spectrum of energy eigenvalues, we now have a discrete spectrum of dipole moments for which such 'bound' states are at all possible. All these conclusions, which are in accord with the virial theorem, reduce to the results of Lévy-Leblond and Balibar obtained from the Heisenberg inequalities ignoring the angular dependence of the dipole potential.