We present the discovery and monitoring observations of Swift 1644+57, a luminous outburst from the nucleus of a galaxy at z = 0.35. Precise astrometry ties the source to within a few hundred parsecs of the nucleus of its host, and suggests a link to the massive black hole that probably resides there. The high luminosity and rapid variability are strongly indicative of a beamed source. We suggest that this event is best explained by the tidal disruption of a passing star by the supermassive black hole, which simultaneously created a powerful panchromatic explosion. However, it has also been proposed that such events may be related to the core collapse of massive stars. Future observations of a sample of similar events, focussing on their locations within the hosts, should distinguish in a straightforward manner between the two proposals.