The number of mRNA and protein molecules expressed from a single gene molecule fluctuates over time. These fluctuations have been attributed, in part, to the random transitioning of promoters between transcriptionally active and inactive states, causing transcription to occur in bursts. However, the molecular basis of transcriptional bursting remains poorly understood. By electron microscopy of single PHO5 gene molecules from yeast, we show that the "activated" promoter assumes alternative nucleosome configurations at steady state, including the maximally repressive, fully nucleosomal, and the maximally non-repressive, nucleosome-free, configuration. We demonstrate that the observed probabilities of promoter nucleosome configurations are obtained from a simple, intrinsically stochastic process of nucleosome assembly, disassembly, and position-specific sliding; and we show that gene expression and promoter nucleosome configuration can be mechanistically coupled, relating promoter nucleosome dynamics and gene expression fluctuations. Together, our findings suggest a structural basis for transcriptional bursting, and offer new insights into the mechanism of transcriptional regulation and the kinetics of promoter nucleosome transitions.