The maximum likelihood estimate of the difference of the means is obtained in sampling from a bivariate normal distribution with unknown variances and co-variance when some of the observations on one of the variables are missing. This estimate is compared to several others by using the mean square error criterion. It is found that, when there are a moderate number of complete pairs of observations, the maximum likelihood estimate has the smallest mean square error for most of the parameter values. Illustrative tables are given to support our conclusions.