  We present Atacama Large Millimeter/submillimeter Array (ALMA) Cycle 2
observations of CO(2-1) emission from the circumnuclear disks in two early-type
galaxies, NGC 1380 and NGC 6861. The disk in each galaxy is highly inclined
($i\,{\sim}\,75^{\circ}$), and the projected velocities of the molecular gas
near the galaxy centers are ${\sim}300\,\mathrm{km \, s^{-1}}$ in NGC 1380 and
${\sim}500\,\mathrm{km \, s^{-1}}$ in NGC 6861. We fit thin disk dynamical
models to the ALMA data cubes to constrain the masses of the central black
holes (BHs). We created host galaxy models using Hubble Space Telescope images
for the extended stellar mass distributions and incorporated a range of
plausible central dust extinction values. For NGC 1380, our best-fit model
yields $M_{\mathrm{BH}} = 1.47 \times 10^8\,M_{\odot}$ with a ${\sim}40\%$
uncertainty. For NGC 6861, the lack of dynamical tracers within the BH's sphere
of influence due to a central hole in the gas distribution precludes a precise
measurement of $M_{\mathrm{BH}}$. However, our model fits require a value for
$M_{\mathrm{BH}}$ in the range of $(1-3) \times 10^9\,M_{\odot}$ in NGC 6861 to
reproduce the observations. The BH masses are generally consistent with
predictions from local BH-host galaxy scaling relations. Systematic
uncertainties associated with dust extinction of the host galaxy light and
choice of host galaxy mass model dominate the error budget of both
measurements. Despite these limitations, the measurements demonstrate ALMA's
ability to provide constraints on BH masses in cases where the BH's projected
radius of influence is marginally resolved or the gas distribution has a
central hole.
