We present $sim0.10^{primeprime}-$resolution Atacama Large Millimeter/submillimeter Array (ALMA) CO(2$-$1) imaging of the arcsecond-scale ($r approx 150$ pc) dusty molecular disk in the giant elliptical galaxy NGC 3258. The data provide unprecedented resolution of cold gas disk kinematics within the dynamical sphere of influence of a supermassive black hole, revealing a quasi-Keplerian central increase in projected rotation speed rising from 280 km s$^{-1}$ at the disks outer edge to $>400$ km s$^{-1}$ near the disk center. We construct dynamical models for the rotating disk and fit beam-smeared model CO line profiles directly to the ALMA data cube. Our models incorporate both flat disks and tilted-ring disks that provide a better fit of the mildly warped structure in NGC 3258. We show that the exceptional angular resolution of the ALMA data makes it possible to infer the host galaxys mass profile within $r=150$ pc solely from the ALMA CO kinematics, without relying on optical or near-infrared imaging data to determine the stellar mass profile. Our model therefore circumvents any uncertainty in the black hole mass that would result from the substantial dust extinction in the galaxys central region. The best model fit yields $M_mathrm{BH} = 2.249times10^9$ $M_odot$ with a statistical model-fitting uncertainty of just 0.18%, and systematic uncertainties of 0.62% from various aspects of the model construction and 12% from uncertainty in the distance to NGC 3258. This observation demonstrates the full potential of ALMA for carrying out highly precise measurements of $M_mathrm{BH}$ in early-type galaxies containing circumnuclear gas disks
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