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In Papers I and II from the Megamaser Cosmology Project, we reported initial observations of H2O masers in an accretion disk of a supermassive black hole at the center of the galaxy UGC 3789, which gave an angular-diameter distance to the galaxy and an estimate of H0 with 16% uncertainty.We have since conducted more very long baseline interferometric observations of the spatial-velocity structure of these H2O masers, as well as continued monitoring of its spectrum to better measure maser accelerations. These more extensive observations, combined with improved modeling of the masers in the accretion disk of the central supermassive black hole, confirm our previous results, but with significantly improved accuracy. We find H0 = 68.9 ± 7.1 km s-1 Mpc-1 this estimate of H0 is independent of other methods and is accurate to ±10%, including sources of systematic error. This places UGC 3789 at an angular-diameter distance of 49.6 ± 5.1 Mpc, with a central supermassive black hole of (1.16 ± 0.12) ×107 M☉. [ABSTRACT FROM AUTHOR]