  As proved by R\'egnier and R\"osler, the number of key comparisons required
by the randomized sorting algorithm QuickSort to sort a list of $n$ distinct
items (keys) satisfies a global distributional limit theorem. Fill and Janson
proved results about the limiting distribution and the rate of convergence, and
used these to prove a result part way towards a corresponding local limit
theorem. In this paper we use a multi-round smoothing technique to prove the
full local limit theorem.
