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Precise star positions near the Sun were measured during the 21 August 2017 total solar eclipse in order to measure their gravitational deflections. The equipment, procedures, and analysis are described in detail. A portable refractor, a CCD camera, and a computerized mount were set up in Wyoming. Detailed calibrations were necessary to improve accuracy and precision. Nighttime measurements taken just before the eclipse provided cubic optical distortion corrections. Calibrations based on star field images 7.4° on both sides of the Sun taken during totality gave linear and quadratic plate constants. A total of 45 images of the sky surrounding the Sun were acquired during the middle part of totality, with an integrated exposure of 22 s. The deflection analysis depended on accurate star positions from the USNO’s UCAC5 star catalog. The final result was a deflection coefficient L  =  1.7512 arcsec, in perfect agreement with the theoretical value, with an uncertainty of only 3%. [ABSTRACT FROM AUTHOR]