Phase fluctuations introduced by the atmosphere are the main limiting factor in attaining diffraction limited performance in extended interferometric arrays at millimeter and submillimeter wavelengths. We report the results of C-PACS, the Combined Array for Research in Millimeter-Wave Astronomy Paired Antenna Calibration System. We present a systematic study of several hundred test observations taken during the 2009-2010 winter observing season where we utilize CARMAs eight 3.5-m antennas to monitor an atmospheric calibrator while simultaneously acquiring science observations with 6.1-m and 10.4-m antennas on baselines ranging from a few hundred meters to ~2 km. We find that C-PACS is systematically successful at improving coherence on long baselines under a variety of atmospheric conditions. We find that the angular separation between the atmospheric calibrator and target source is the most important consideration, with consistently successful phase correction at CARMA requiring a suitable calibrator located $lesssim$6$^circ$ away from the science target. We show that cloud cover does not affect the success of C-PACS. We demonstrate C-PACS in typical use by applying it to the observations of the nearby very luminous infrared galaxy Arp 193 in $^{12}$CO(2-1) at a linear resolution of ~70 pc (0.12 x 0.18), 3 times better than previously published molecular maps of this galaxy. We resolve the molecular disk rotation kinematics and the molecular gas distribution and measure the gas surface densities and masses on 90 pc scales. We find that molecular gas constitutes $sim30%$ of the dynamical mass in the inner 700 pc of this object with a surface density $sim10^4 M_odot$ pc$^{-2}$; we compare these properties to those of the starburst region of NGC 253.
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