Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group $G$ of order $n$ satisfying the Hall--Paige condition is $(e^{-1/2} + o(1)) \, |G^\text{ab}| \, n!^2/n^n$.