M104 (NGC 4594; the Sombrero galaxy) is a nearby, well-studied elliptical galaxy included in scores of surveys focused on understanding the details of galaxy evolution. Despite the importance of observations of M104, a consensus distance has not yet been established. Here, we use newly obtained Hubble Space Telescope optical imaging to measure the distance to M104 based on the tip of the red giant branch method. Our measurement yields the distance to M104 to be 9.55 +/- 0.13 +/- 0.31 Mpc equivalent to a distance modulus of 29.90 +/- 0.03 +/- 0.07 mag. Our distance is an improvement over previous results as we use a well-calibrated, stable distance indicator, precision photometry in a optimally selected field of view, and a Bayesian Maximum Likelihood technique that reduces measurement uncertainties. The most discrepant previous results are due to Tully-Fisher method distances, which are likely inappropriate for M104 given its peculiar morphology and structure. Our results are part of a larger program to measure accurate distances to a sample of well-known spiral galaxies (including M51, M74, and M63) using the tip of the red giant branch method.