MVUE for the mean with one observation: normal with same mean and variance

Based on one observation X from a N(., .) distribution where .(> 0) is an unknown parameter, this article provides an exact expression of the minimum variance unbiased estimator of . based on the complete and sufficient statistic |X|. Even though N(., .) belongs to a one-parameter exponential family, I discuss some small surprises and intricacies that are worth noting.