2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/202216We match the vacuum, stationary, cylindrically symmetric solution of Einstein's field equations with $Λ$, in a form recently given by Santos, as an exterior to an infinite cylinder of dust cut out of a Gödel universe. There are three cases, depending on the radius of the cylinder. Closed timelike curves are present in the exteriors of some of the solutions. There is a considerable similarity between the spacetimes investigated here and those of van Stockum referring to an infinite cylinder of rotating dust matched to vacuum, with $Λ=0$.11 pages, LaTeX 2.09, no figures. Submitted to Classical and Quantum GravityGeneral Relativity and Quantum Cosmologytext