We present Atacama Large Millimeter/submillimeter Array (ALMA) Cycle 2 observations of CO(21) emission from the circumnuclear disks in two early-type galaxies, NGC 1380 and NGC 6861. The disk in each galaxy is highly inclined (
i75), and the projected velocities of the molecular gas near the galaxy centers are 300 km s1 in NGC 1380 and 500 km s1 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 MBH = 1.47 108 Mwith a 40% uncertainty. For NGC 6861, the lack of dynamical tracers within the BHs sphere of influence due to a central hole in the gas distribution precludes a precise measurement of MBH. However, our model fits require a value for MBH in the range of (13) 109 Min NGC 6861 to reproduce the observations. The BH masses are generally consistent with predictions from local BHhost 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 ALMAs ability to provide constraints on BH masses in cases where the BHs projected radius of influence is marginally resolved or the gas distribution has a central hole.