We consider the Josephson-vortex motion in a long one-dimensional Jose
phson junction in a thin film. We show that this Josephson vortex is s
imilar to a mesoscopic capacitor. We demonstrate that a single-Cooper-
pair tunnelling results in non-linear Bloch-type oscillations of a Jos
ephson vortex in a current-biased Josephson junction. We find the freq
uency and the amplitude of this motion.