Determining the distribution of population extinction times is a fundamental problem in theoretical population biology. In particular, the tail properties, patterns in the probability of long-term persistence, have not been studied. Further, until now there have been no experimental or observational data sets with which to empirically test the "rare event" predictions of the standard stochastic theory of extinction, which holds that extinction times should be exponentially distributed. I performed an experimental study of extinction in a large number of replicate (n = 1076) laboratory populations of the waterflea Daphnia pulicaria. Observed extinction time ranged from 1 to 1239 days. Statistical models supported the hypothesis of a power-law distribution over the exponential distribution and other alternatives. This pattern contradicts the notion that population extinction time has an exponential tail, questioning its ubiquitous use in theoretical ecology. It is also a rare instance of a data set that exhibits power-law scaling under appropriate statistical criteria.