The power spectrum of a stationary Gaussian random process is estimated when partial knowledge of the autocorrelation function is available {\em a priori}. Particular attention is paid to the case when the {\em a priori} knowledge is not precise, i.e., when there are errors in the measurements, perhaps due to the presence of noise. In the special case when the {\em a priori} knowledge consists ofnpoints of the autocorrelation function, Burg's method of picking the spectrum which maximizes the entropy of the Gaussian process has been recently extended by Newman to account for a weighted average error in the estimates of the correlation function points. A new method is suggested here that uses the mutual information principle (MIP) of Tzannes and Noonan. The firstnpoints of the correlation function (obtained with errors) are used to derive an approximate spectrum by Burg's or any other method. This spectrum, as well as the error constraints involved, is then used to arrive at the underlying spectrum in the framework of the MIP approach.