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Accurate estimation of the amplitude and frequency parameters of sinusoidal signals from noisy observations is an important problem in many signal processing applications. In this paper, the problem is investigated under the assumption of non-Gaussian noise in general and Laplace noise in particular. It is proven mathematically that the maximum likelihood estimator derived under the condition of Laplace white noise is able to attain an asymptotic Cramer-Rao lower bound which is one half of that achieved by periodogram maximization and nonlinear least squares. It is also proven that when applied to non-Laplace situations, the Laplace maximum likelihood estimator, which may also be referred to as the nonlinear least-absolute-deviations estimator, can achieve an even higher statistical efficiency especially when the noise distribution has heavy tails. A computational procedure is proposed to overcome the difficulty of local extrema in the likelihood function. Simulation results are provided to validate the analytical findings.