A relic particle X which annihilates with its antiparticle [Formula: see text] into electromagnetically-interacting particles, e.g. [Formula: see text], γγ, etc., will lead to a distortion in the spectrum of the cosmic microwave background (CMB). The magnitude of this effect is calculated and compared with known limits on the deviations of the CMB from a Planck spectrum. Limits which can be placed on such a particle are found to be competitive with similar limits from Big Bang nucleosynthesis.