Optimal Doppler velocity estimation is explored for a standard Gaussian signal measurement model and thematic maximum likelihood (ML) and Bayes estimation. Because the model considered depends on a vector parameter (velocity, spectrum width (SW), and signal-to-noise ratio (SNR), the exact formulation of an ML or Bayes solution involves a system of coupled equations which cannot be made explicit for any of the parameters. In the past, iterative methods have been suggested for solving the required equations. In addition to being computationally intensive, it is unclear whether an iterative method can be constructed to converge well under general conditions. Simple computational forms are shown to exist when SW and SNR are assumed known. An information theoretic concept is used to propose an adaptive extension of these equations to the general case of SW and SNR unknown. This new idea is developed to the poise of operational application. A Monte Carlo simulations experiment is used to verify that the method can work; the example presented considers the particularly difficult situation of no a priori information for either SW or SNR under the additional constraint of a very small (20 pulse samples) sample size. The improved performance of this new Doppler velocity estimator is documented by comparison with derived optimal bounds and with the performance of the well-known pulse pair (PP) method. Small-sample estimator statistics are presented; and Bayes estimator results, assuming known SW and SNR, are used to provide true performance bounds for comparison. Cramer-Rao (CR) bounds are also derived and shown to be inferior to the Bayes bounds in the small sample case considered.