We have observed profound variability in the radio flux density of the quasar PKS 0405-385 on timescales of less than an hour; this is unprecedented amongst extragalactic sources. If intrinsic to the source, these variations would imply a brightness temperature 10^21 K, some nine orders of magnitude larger than the inverse Compton limit for a static synchrotron source, and still a million times greater than can be accommodated with bulk relativistic motion at a Lorentz factor equal to 10. The variability is intermittent with episodes lasting a few weeks to months. Our data can be explained most sensibly as interstellar scintillation of a source component which is < 5 microarcsec in size - a source size which implies a brightness temperature > 5 times 10^14 K, still far above the inverse Compton limit. Simply interpreted as a steady, relativistically beamed synchrotron source, this would imply a bulk Lorentz factor 1000.
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