Graphene's remarkable mechanical and electrical properties, combined with its compatibility with existing planar silicon-based technology, make it an attractive material for novel computing devices. We report the development of a nonvolatile memory element based on graphene break junctions. Our devices have demonstrated thousands of writing cycles and long retention times. We propose a model for device operation based on the formation and breaking of carbon atomic chains that bridge the junctions. We demonstrate information storage based on the concept of rank coding, in which information is stored in the relative conductance of graphene switches in a memory cell.