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 1.1 \times 10 ^ { -24 } { W } / { \sqrt { Hz } } in the band of operation for an integration time of 1.8 \times 10 ^ { 3 } s .