The discreteness of charge in units of e led Schottky in 1918 to predict that the electrical current in a vacuum tube fluctuates even if all spurious noise sources are eliminated carefully. This phenomenon is now widely known as shot noise. In recent years, shot noise in mesoscopic conductors, where charge motion is quantum-coherent over distances comparable to the system size, has been studied extensively. In those experiments, charge does not propagate as an isolated entity through free space, as for vacuum tubes, but is part of a degenerate and quantum-coherent Fermi sea of charges. It has been predicted that shot noise in mesoscopic conductors can disappear altogether when the system is tuned to a regime where electron motion becomes classically chaotic. Here we experimentally verify this prediction by using chaotic cavities where the time that electrons dwell inside can be tuned. Shot noise is present for large dwell times, where the electron motion through the cavity is 'smeared' by quantum scattering, and it disappears for short dwell times, when the motion becomes classically deterministic.