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In this paper, a new structured approach for obtaining uniformly best non-Bayesian biased estimators, which attain minimum-mean-square-error performance at any point in the parameter space, is established. We show that if a uniformly best biased (UBB) estimator exists, then it is unique, and it can be directly obtained from any locally best biased (LBB) estimator. A necessary and sufficient condition for the existence of a UBB estimator is derived. It is shown that if there exists an optimal bias, such that this condition is satisfied, then it is unique, and its closed-form expression is obtained. The proposed approach is exemplified in two nonlinear estimation problems, where uniformly minimum-variance-unbiased estimators do not exist. In the considered examples, we show that the UBB estimators outperform the corresponding maximum-likelihood estimators in the MSE sense.