Accurate measurement of RNA transcripts from single cells will enable the precise classification of cell types and characterization of the heterogeneity in cell populations that play key roles in normal cellular physiology and diseases. As a step towards this end, we have developed a microfluidic device and methods for automatic hydrodynamic capture of single mammalian cells and subsequent immobilization and digital counting of polyadenylated mRNA molecules released from the individual cells. Using single-molecule fluorescence imaging, we have demonstrated that polyadenylated mRNA molecules from single HeLa cells can be captured within minutes by hybridization to polydeoxyribothymidine oligonucleotides covalently attached on the glass surface in the device. The total mRNA molecule counts in the individual HeLa cells are found to vary significantly from one another. Our technology opens up the possibility of direct digital enumeration of RNA transcripts from single cells with single-molecule sensitivity using a single integrated microfluidic device.