The problems have long been recognized in supporting a conventional, disc-shaped optical element in the presence of gravity, so as to minimize the gravity-induced deflections. With the increasing size and quality of optics that are being used in space come correspondingly difficult problems in simulating a zero-gravity situation while supporting the optic on Earth. These problems are made worse by the fact that no additional stresses and strains should be introduced into the optic as it slowly changes shape because of, for instance, temperature drifts. In other words, a zero-gravity support should be kinematic. In this paper, we present a scheme for a zero-gravity support for a cylindrical optic. The type of optic being considered is used in grazing incidence X-ray telescopes, and consists of a cylindrical shell with a reflecting surface on the inside of the cylinder. The zero-gravity support presented permits extremely small gravity-induced deflections, yet contacts the mirror in only four places, and is therefore rather easily made kinematic. Moreover, no counterbalances or calibrated forces are needed - the distribution of the weight of the mirror itself guarantees that the off-loading will be calibrated correctly. We discuss the application of this concept for the alignment of an X-ray telescope. Using some thin-member approximations, we give some analytical approximations of the deflections. Also, some results of a finite element method analysis of the support are given. These results are interpreted in terms of a previously published set of functions that are orthonormal over the surface of a cylinder.