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

Based on one observation X from a N([theta], [theta]) distribution where [theta](> 0) is an unknown parameter, this article provides an exact expression of the minimum variance unbiased estimator of [theta] based on the complete and sufficient statistic |X|. Even though N([theta], [theta]) belongs to a one-parameter exponential family, I discuss some small surprises and intricacies that are worth noting. KEY WORDS: Maximum likelihood estimator; Minimum variance; Rao-Blackwellization; Unbiased estimation.