The Axion Dark Matter eXperiment (ADMX) was designed to detect ultra-weakly interacting relic axion particles by searching for their conversion to microwave photons in a resonant cavity positioned in a strong magnetic field. Given the extremely low expected axion-photon conversion power we have designed, built and operated a microwave receiver based on a Superconducting QUantum Interference Device (SQUID). We describe the ADMX receiver in detail as well as the analysis of narrow band microwave signals. We demonstrate the sustained use of a SQUID amplifier operating between 812 and 860 MHz with a noise temperature of 1 K. The receiver has a noise equivalent power of 1.1x10^-24 W/sqrt(Hz) in the band of operation for an integration time of 1.8x10^3 s.
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