Tight bounds on the minimum mean square error for the additive Gaussian noise channel are derived, when the input distribution is constrained to be epsilon-close to a Gaussian reference distribution in terms of the Kullback--Leibler divergence. The distributions that attain the bounds are shown be Gaussian whose means are identical to that of the reference distribution and whose covariance matrices are defined implicitly via systems of matrix equations. The estimator that attains the upper bound is identified as a minimax optimal estimator that is robust against deviations from the assumed prior. The lower bound is shown to provide a potentially tighter alternative to the Cramer--Rao bound. Both properties are illustrated with numerical examples.