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A generalized Burg technique has been developed recently by Burg, Luenberger, and Wenger for maximum likelihood estimation of structured covariance matrices. Uniqueness of the estimate in the restricted subset of the class of nonnegative definite symmetric matrices is not known. In this paper it is shown that the positive definite estimate over the class of nonnegative definite, doubly symmetric (symmetric about both the main and minor diagonals) matrices is unique. Moreover, if the minor diagonal symmetrized version of the sample covariance matrix is nonsingular, it is the unique estimate. If the minor diagonal symmetrized sample covariance matrix is singular and a positive definite estimate exists, then the estimate is unique.