We report an experiment in which an optical vortex is stored in a vapor of Rb atoms. Because of its 2pi phase twist, this mode, also known as the Laguerre-Gauss mode, is topologically stable and cannot unwind even under conditions of strong diffusion. For comparison, we stored a Gaussian beam with a dark center and a uniform phase. Contrary to the optical vortex, which stays stable for over 100 micros, the dark center in the retrieved flat-phased image was filled with light after a storage time as short as 10 micros. The experiment proves that higher electromagnetic modes can be converted into atomic coherences and that modes with phase singularities are robust to decoherence effects such as diffusion. This opens the possibility to more elaborate schemes for classical and quantum information storage in atomic vapors.