In Papers I and II from the Megamaser Cosmology Project , we reported initial observations of H _ { 2 } O 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 H _ { 0 } with 16 % uncertainty . We have since conducted more VLBI observations of the spatial-velocity structure of these H _ { 2 } O 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 { H _ { 0 } } = 68.9 \pm 7.1 km s ^ { -1 } Mpc ^ { -1 } ; this estimate of H _ { 0 } is independent of other methods and is accurate to \pm 10 % , including sources of systematic error . This places UGC 3789 at an angular-diameter distance of 49.6 \pm 5.1  Mpc , with a central supermassive black hole of ( 1.16 \pm 0.12 ) \times 10 ^ { 7 }  M _ { \odot } .