The regular series of peaks observed in the spectrum of the diffuse interstellar absorption band at 6614Angstroms is interpreted in terms of an isotope shift. It is supposed that the band is an electronic transition of some carbon-rich molecular species in the gaseous phase; that the electronic transition is accompanied by a simultaneous change in the vibrational state; and that the isotope shift is inertial, the consequence of the dependence of the vibrational frequency on the extent to which the more abundant ^12C atoms in the molecule are replaced by the more massive ^13C. A model is presented in which each isotopic variety of the molecule gives rise to a narrow absorption feature, the central wavelength of which is shifted by an amount proportional to the number of ^13C atoms in the molecule, and the intensity of which is given by the Poisson law governing the relative abundance of different isotopic varieties. The resulting synthetic spectrum resembles the observed spectrum most closely when the parameters of the model are given values appropriate to a molecule containing something like 30-100 carbon atoms. The sharpness of the peaks in the spectrum of lambda6614 implies that the molecule may have a high symmetry, of a kind to be found in cyclic polyynes, in fullerenes such as the soccer-ball molecule C_60 and in some of the derivatives such as the fulleranes C_60H_m, but not in linear polyynes or in planar molecules such as polycyclic aromatic hydrocarbons. The isotope shift is capable of producing a variety of band profiles, but gives rise to asymmetric profiles with extended blue wings only when the molecule is already vibrating at the time of the absorption event; this condition seems unlikely to be met at the low temperature of the interstellar medium, so the observed scarcity of narrow diffuse bands with this kind of asymmetry may have a simple explanation. The diffuseness of lambda6614 and most of the other narrow bands may owe more to the shading caused by the isotope shift than to that caused by molecular rotation or internal conversion.