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This paper introduces a maximum-likelihood method for the nonparametric estimation of smooth spectra from irregularly sampled observations, which is abbreviated as LIMES (Likelihood-based Method for Estimation of Spectra). As a byproduct, LIMES also provides an estimate of the data covariance matrix that may be of interest in its own right. Spectral estimation from irregularly sampled data is a rather difficult problem and there are only a handful of methods in the literature that can be used for such a task. Of these already existing methods we consider the Daniell method (DAM) for comparison with LIMES. Computationally, LIMES is more complex than DAM. On the other hand, DAM is much less accurate than LIMES in the irregularly sampled data case and for spectra with a relatively large bandwidth. In a nutshell, LIMES should be the method of choice in the unevenly sampled data applications that require high statistical performance and can tolerate an increased computational burden.