The supermassive black hole of M87 is one of the most massive black holes known and has been the subject of several stellar and gas-dynamical mass measurements; however the most recent revision to the stellar-dynamical black hole mass measurement is a factor of about two larger than the previous gas-dynamical determinations. Here, we apply comprehensive gas-dynamical models that include the propagation of emission-line profiles through the telescope and spectrograph optics to new Space Telescope Imaging Spectrograph observations from the Hubble Space Telescope. Unlike the previous gas-dynamical studies of M87, we map out the complete kinematic structure of the emission-line disk within about 40 pc from the nucleus, and find that a small amount of velocity dispersion internal to the gas disk is required to match the observed line widths. We examine a scenario in which the intrinsic velocity dispersion provides dynamical support to the disk, and determine that the inferred black hole mass increases by only 6%. Incorporating this effect into the error budget, we ultimately measure a mass of M_BH = (3.5^{+0.9}_{-0.7}) x 10^9 M_sun (68% confidence). Our gas-dynamical black hole mass continues to differ from the most recent stellar-dynamical mass by a factor of two, underscoring the need for carrying out more cross-checks between the two main black hole mass measurement methods. ; Comment: 11 pages, 7 figures, accepted for publication in ApJ