This paper studies the design of an optimal privacy-aware estimator of a public random variable based on noisy measurements which contain private information. The public variable carries also non-private information, but, its estimate will be correlated with the private information due to the estimation process. The objective is to design an optimal estimator of the public random variable such that the leakage of private information, via the estimation process, is kept below a certain level. The privacy metric is defined as the discrete conditional entropy of the private variable given the output of the estimator. We show that the optimal privacy-aware estimator is the solution of a (possibly infinite-dimensional) convex optimization problem when the estimator has access to either the measurement or the measurement together with the private information. We study the optimal perfect-privacy estimation problem that ensures the estimate of the public variable is independent of the private information. A necessary and sufficient condition is derived guaranteeing that an estimator satisfies the perfect-privacy requirement.