For a bivariate population, with a probability density $p(|x - \theta|)$ where $p(t)$ is strictly decreasing for $t \geqq 0$, when a single observation is taken, the usual confidence sets are circles of constant radius centered at the observed value. Previous results of Kiefer imply that these circles have the minimax property of minimizing the maximum expected Lebesgue measure of the confidence sets for a given lower confidence level. It is now shown here that subject to mild conditions on $p(t)$, the confidence circles are also essentially unique in having the minimax property. The result generalizes the result proved previously for the bivariate normal case.