We derive the elastic collision rate for a harmonically trapped Fermi gas in the extreme unitarity limit where the s-wave scattering cross section is [Formula Presented] with [Formula Presented] the relative momentum. The collision rate is given in the form [Formula Presented]—the product of a universal collision rate [Formula Presented] and a dimensionless function of the ratio of the temperature T to the Fermi temperature [Formula Presented] We find that I has a peak value of [Formula Presented] at [Formula Presented] [Formula Presented] for [Formula Presented] and [Formula Presented] for [Formula Presented] We estimate the collision rate for recent experiments on a strongly-interacting degenerate Fermi gas of atoms. © 2003 The American Physical Society.