Carbon nanotubes have potential in the development of high-speed and power-efficient logic applications(1-7). However, for such technologies to be viable, a high density of semiconducting nanotubes must be placed at precise locations on a substrate. Here, we show that ion-exchange chemistry can be used to fabricate arrays of individually positioned carbon nanotubes with a density as high as 1x10(9) cm(-2)-two orders of magnitude higher than previous reports(8,9). With this approach, we assembled a high density of carbon-nanotube transistors in a conventional semiconductor fabrication line and then electrically tested more than 10,000 devices in a single chip. The ability to characterize such large distributions of nanotube devices is crucial for analysing transistor performance, yield and semiconducting nanotube purity.