Since the time of the Voyager flybys of Saturn in 1980-1981, Saturn's eccentric F ring has been known to be accompanied on either side by faint strands of material. New Cassini observations show that these strands, initially interpreted as concentric ring segments, are in fact connected and form a single one-arm trailing spiral winding at least three times around Saturn. The spiral rotates around Saturn with the orbital motion of its constituent particles. This structure is likely the result of differential orbital motion stretching an initial cloud of particles scattered from the dense core of the F ring. Different scenarios of formation, implying ringlet-satellite interactions, are explored. A recently discovered moon candidate, S/2004 S6, is on an orbit that crosses the F-ring core at the intersection of the spiral with the ring, which suggests a dynamical connection between S/2004 S6 and the spiral.