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In this paper, a minimax mean-squared error (MSE) estimator is developed for estimating an unknown deterministic parameter vector in a linear model, subject to noise covariance uncertainties. The estimator is designed to minimize the worst-case MSE across all norm-bounded parameter vectors, and all noise covariance matrices, in a given region of uncertainty. The minimax estimator is shown to have the same form as the estimator that minimizes the worst-case MSE over all norm-bounded vectors for a least-favorable choice of the noise covariance matrix. An example demonstrating the performance advantage of the minimax MSE approach over the least-squares and weighted least-squares methods is presented.