In arena experiments with the walking fruit fly, we found a remarkable persistence of orientation toward a landmark that disappeared during the fly's approach. The directional stability achieved by 'after-fixation' allows a fly to continue pursuit under natural conditions, where a selected target is frequently concealed by surrounding structures. The persistence of after-fixation was investigated in Buridan's paradigm, where a fly walks persistently back and forth between two inaccessible landmarks. Upon disappearance of a selected target, the flies maintained their intended course for more than 15 body lengths of approximately 2.5 mm in about 50% of the trials. About 13% even exceeded 75 body lengths. About 88% of the approaches clustered in equal portions around peaks at 2.4 s and 8.6 s. About 12% of the approaches persisted even longer. In contrast, a single peak at about 2.2 s is sufficient to describe the persistence of orientation in a random walk. The ability to pursue an invisible landmark is disturbed neither by a transient angular deviation from the course toward this landmark, when this target disappeared, nor by a distracting second landmark. Accordingly, after-fixation seems to require an internal representation of the direction toward the concealed target, and idiothetical course control to maintain this direction.