The univariate generalized extreme value (GEV) distribution is the most commonly used tool for analyzing the properties of rare events. The ever greater utilization of Bayesian methods for extreme value analysis warrants detailed theoretical investigation, which has thus far been underdeveloped. Even the most basic asymptotic results are difficult to obtain because the GEV fails to satisfy standard regularity conditions. Here, we prove that the posterior distribution of the GEV parameter vector, given $n$ independent and identically distributed samples, converges in distribution to a trivariate normal distribution. The proof necessitates analyzing integrals of the GEV likelihood function over the entire parameter space, which requires considerable care because the support of the GEV density depends on the parameters in complicated ways.