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We report the successful operation of a superconductor-insulator-superconductor (SIS) finline mixer operating near the superconducting energy gap of Nb. The mixer employs a new type of Pickett-Potter horn-reflector (PPHR) antenna, which exhibits low sidelobes and low cross-polarization levels, and yet is easy to fabricate. The SIS tunnel junction and all of the integrated superconducting tuning circuits are fabricated from Nb using planar-circuit technology. The mixer employs an antipodal finline section, deposited on one side of a quartz substrate, which transforms the high impedance of the waveguide ([approximately equal to] 300 [ohm]) to the low impedance of the microstrip line ([approximately equal to] 20 [ohm]). The Nb/Al-oxide/Nb tunnel junction is fabricated at the same time as the finline circuit. In this paper, we describe the design and testing of the mixer, and pay particular attention to the electromagnetic design of the PPHR antenna. We investigate the noise temperature and gain of the mixer over 642-714 GHz, and analyze the experimental results using rigorous theories that were developed specifically for the purpose. Our investigation demonstrates that finline mixers have good performance both below and above the superconducting energy gap. Index Terms--Finline transitions, horn antennas, submillimeter-wave receivers, superconductor-insulator-superconductor (SIS) mixers.