Radiometer spectrometers are used in millimeter-wave radio astronomy for the spectral measurement of molecular rotational transitions. The spectrum of interest spans 10s of GHz and the measurement time is large in order to obtain useful signal-to-noise ratio. We describe here the design and performance of a multichannel acousto-optic spectrometer for radio astronomy spectroscopy applications. This instrument has 4 channels and can process signals from 4 separate sources simultaneously. The bandwidth of each channel is 1 GHz and the frequency resolution is 1 MHz, providing simultaneous processing over 4,000 1 MHz channels.