We present an analytical theory for the operation of a Cerenkov free-electron laser which includes diffraction of the optical mode in the direction transverse to the electron beam. Because the width of the optical mode depends on gain, the usual cubic dispersion relation is replaced by a 5/2-power dispersion relation, which allows two roots. These roots both have positive real parts, indicating that they are slow waves. For a narrow electron beam, the optical mode is much wider than the beam, thus reducing the gain by an order of magnitude from that predicted by the two-dimensional theory. In the limit of a wide electron beam, the two-dimensional theory is recovered.