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 $H_0 = 73.9 pm 3.0$ 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 ${pm}1{sigma}$, 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 95-99% for different treatments of the peculiar velocities.
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