The next-generation very-high-energy (VHE) gamma-ray observatory, the Cherenkov Telescope Array, will feature dozens of imaging atmospheric Cherenkov telescopes (IACTs), each with thousands of pixels of photo-sensors. To be affordable and reliable, reading out such an array of up to a million channels requires event recording technology that is highly integrated and modular, with a low cost per channel. We present the design and performance of a chip targeted to this application: the TeV Array Readout with GSa/s sampling and Event Trigger (TARGET). This application-specific integrated circuit (ASIC) has 16 parallel input channels, a 4096-sample buffer for each channel, adjustable input termination, self-trigger functionality, and tight window-selected readout. We report the performance of the first-generation version of this chip (TARGET 1) in terms of sampling frequency, power consumption, dynamic range, current-mode gain, analog bandwidth, and cross talk. The large number of channels per chip allows a low cost per channel ($10 to $20 including front-end and back-end electronics but not including photosensors) to be achieved with a TARGET-based IACT readout system. In addition to basic performance parameters of the TARGET 1 chip, we present a camera module prototype as well as a second-generation chip (TARGET 2), both of which have now been produced.