We demonstrate microscopic fluidic control and memory elements through the use of an aqueous viscoelastic polymer solution as a working fluid. By exploiting the fluid's non-Newtonian rheological properties, we were able to demonstrate both a flux stabilizer and a bistable flip-flop memory. These circuit elements are analogous to their solid-state electronic counterparts and could be used as components of control systems for integrated microfluidic devices. Such miniaturized fluidic circuits are insensitive to electromagnetic interference and may also find medical applications for implanted drug-delivery devices.