The high-resolution frequency estimators most commonly used, such as MUSIC, ESPRIT and Yule-Walker, determine estimates of the sinusoidal frequencies from the sample covariances of noise-corrupted data. In this paper, a frequency estimation method termed Approximate Maximum Likelihood (AML) is derived from the approximate likelihood function of sample covariances. The statistical performance of AML is studied, both analytically and numerically, and compared with the Cramer-Rao bound as well as the statistical performance corresponding to the aforementioned methods of frequency estimation. AML is shown to provide the minimum asymptotic error variance in the class of all estimators based on a given set of covariances. The implementation of the AML frequency estimator is discussed in detail. The paper also introduces an AML-based procedure for estimating the number of sinusoidal signals in the measured data, which is shown to possess high detection performance.