We report a study of the influence of molecular fluctuations on a limit-cycle model of circadian rhythms based on the regulatory network of a gene involved in a biochemical clock. The molecular fluctuations may become important because of the low number of molecules involved in such genetic regulatory networks at the subcellular level. The molecular fluctuations are described by a birth-and-death stochastic process ruled by the chemical master equation of Nicolis and co-workers and simulated by Gillespie's algorithm. The robustness of the oscillations is characterized, in particular, by the probability distribution of the first-return times and the autocorrelation functions of the noisy oscillations. The half-life of the autocorrelation functions is studied as a function of the size of the system which controls the magnitude of the molecular fluctuations and of the degree of cooperativity of some reaction steps of the biochemical clock. The role of the attractivity of the limit cycle is also discussed.