A method is described for computing the asymptotic variance of the maximum likelihood correlation estimator $\tilde{\rho}$, which uses the number of observations in symmetrically placed corner regions, ignoring a middle section. It is shown that the optimum location of the regions varies with the true value of the correlation. Tables and figures are presented showing the optimal locations and the efficiency of the estimation procedure for various locations under two conditions: one, where cost depends on total sample size and two, where cost depends on the number of observations in the corner regions. In the second case, the method discussed is shown to be able to attain any given precision more cheaply than estimation by the product moment correlation coefficient.