We demonstrate the first rewritable memory in thermally drawn fibers. A high tellurium-content chalcogenide glass, contacted by metallic electrodes internal to the fiber structure, is drawn from a macroscopic preform. An externally applied voltage is utilized to switch between a high resistance (OFF) and a low resistance (ON) state; this in turn allows the fibers to function as a memory device reminiscent of the ovonic switch. The difference between the ON and OFF states is found to be four orders of magnitude. The glass-crystal phase transition is localized to micrometer-wide filaments, whose position can be optically controlled along the fiber axis. An architecture that enabled the encoding of multiple bits per fiber is described.