  A family of exact solutions is presented which represents a rigidly rotating
cylinder of dust in a background with a negative cosmological constant. The
interior of the infinite cylinder is described by the Godel solution. An exact
solution for the exterior solution is found which depends both on the rotation
of the interior and on its radius. For values of these parameters less than a
certain limit, the exterior solution is shown to be locally isomorphic to the
Linet-Tian solution. For values larger than another limit, it is shown that the
exterior solution extends into a region which contains closed timelike curves.
At large distances from the source, the space-time is shown to be asymptotic
locally to anti-de Sitter space.
