We consider the problem of jointly optimum modulation and estimation of a real-valued random parameter, conveyed over an AWGN channel, where the performance metric is the exponential decay rate of the probability that the estimation error would exceed a certain threshold. Our basic result is in providing an exact characterization of the fastest achievable exponential decay rate, among all possible modulator-estimator (transmitter-receiver) pairs, where the modulator is limited only in the signal power. This exponential rate turns out to be given by the reliability function of the AWGN channel. We also discuss several ways to achieve this performance, as well as several (rather surprising) observations, modifications and extensions.