We present a measurement of the Hubble constant made using geometric distance measurements to megamaser-hosting galaxies. We have applied an improved approach for fitting maser data and obtained better distance estimates for four galaxies previously published by the Megamaser Cosmology Project: UGC 3789, NGC 6264, NGC 6323, and NGC 5765b. Combining these updated distance measurements with those for the maser galaxies CGCG 074-064 and NGC 4258, and assuming a fixed velocity uncertainty of 250 km s(-1) associated with peculiar motions, we constrain the Hubble constant to be H-0 = 73.9 +/- 3.0 km s(-1) Mpc(-1) independent of distance ladders and the cosmic microwave background. This best value relies solely on maser-based distance and velocity measurements, and it does not use any peculiar velocity corrections. Different approaches for correcting peculiar velocities do not modify H-0 by more than +/- 1 sigma, with the full range of best-fit Hubble constant values spanning 71.8-76.9 km s(-1) Mpc(-1). We corroborate prior indications that the local value of H-0 exceeds the early-universe value, with a confidence level varying from 95% to 99% for different treatments of the peculiar velocities.