Circadian oscillators, which provide internal daily periodicity, are found in a variety of living organisms, including mammals, insects, plants, fungi and cyanobacteria. Remarkably, these biochemical oscillators are resilient to external and internal modifications, such as temperature and cell division cycles. They have to be 'fluctuation (noise) resistant' because relative fluctuations in the number of messenger RNA and protein molecules forming the intracellular oscillators are likely to be large. In multicellular organisms, the strong temporal stability of circadian clocks, despite molecular fluctuations, can easily be explained by intercellular interactions. Here we study circadian rhythms and their stability in unicellular cyanobacteria Synechoccocus elongatus. Low-light-level microscopy has allowed us to measure gene expression under circadian control in single bacteria, showing that the circadian clock is indeed a property of individual cells. Our measurements show that the oscillators have a strong temporal stability with a correlation time of several months. In contrast to many circadian clocks in multicellular organisms, this stability seems to be ensured by the intracellular biochemical network, because the interactions between oscillators seem to be negligible.