The harmonic decomposition of a random process into a sum of sinusoids in white noise is an important problem with applications in a number of different areas. As a result of the work of V. F. Pisarenko, it has been shown that the sinusoidal frequencies and the white noise power are determined by the minimum eigenvalue and the corresponding eigenvector of the autocorrelation matrix. In this paper, an efficient algorithm is presented for finding this eigenvalue and eigenvector. In addition to its being computationally more efficient than the power method, it has a "built-in" criterion for selecting the model order to use in the decomposition. Some examples are presented and the results are compared to those obtained using other approaches.