Protein levels have been shown to vary substantially between individual cells in clonal populations. In prokaryotes, the contribution to such fluctuations from the inherent randomness of gene expression has largely been attributed to having just a few transcripts of the corresponding mRNAs. By contrast, eukaryotic studies tend to emphasize chromatin remodeling and burst-like transcription. Here, we study single-cell transcription in Escherichia coli by measuring mRNA levels in individual living cells. The results directly demonstrate transcriptional bursting, similar to that indirectly inferred for eukaryotes. We also measure mRNA partitioning at cell division and correlate mRNA and protein levels in single cells. Partitioning is approximately binomial, and mRNA-protein correlations are weaker earlier in the cell cycle, where cell division has recently randomized the relative concentrations. Our methods further extend protein-based approaches by counting the integer-valued number of transcript with single-molecule resolution. This greatly facilitates kinetic interpretations in terms of the integer-valued random processes that produce the fluctuations.