We have measured the properties of a superconductor‐insulator‐superconductor quasiparticle mixer which is operated in the quantum limit. Single sideband conversion gain larger than +4 dB was observed at 36 GHz with a mixer noise temperatureT= 9±6 K, which is to be compared with the (Planck) quantum limit h&slash;&ohgr;/(k ln 2)≃2.5 K. Complete three‐port mixer calculations are presented which are in good agreement with the gain measurements. The dynamic resistance was observed to become infinite and then negative as the source conductance was decreased. This implies that arbitrarily large gain is available. The negative resistance is accompanied by IF oscillations.