Let $x( t )$ be a diffusion resulting from the stochastic perturbation of a deterministic dynamical system by a nondegenerate white noise. Let $\tau $ be the time of first exit of $x( t )$ from a domain on which the deterministic flow has a single simple attracting critical point and is inward at the boundary. Previous results on determining the statistics of $\tau $ include the asymptotic behavior of the first moment and certain decay rates of probabilities of containment past $t = T$ as the strength of the noise tends to zero. In this work the actual asymptotic distribution of $\tau $ in this limit is determined to be exponential in the potential case. The singularly perturbed equations describing this limit exhibit Ackerberg–O’Malley resonance.