We present Atacama Large Millimeter/submillimeter Array (ALMA) observations of 1.1 mm dust continuum and CO 2-1 emission toward six dense cores within the Ophiuchus molecular cloud. We detect compact, sub-arcsecond continuum structures toward three targets, two of which (Oph A N6 and SM1) are located in the Ophiuchus A ridge. Two targets, SM1 and GSS 30, contain two compact sources within the ALMA primary beam. We argue that several of the compact structures are small ($R lesssim 80$ au) accretion disks around young protostars, due to their resolved, elongated structures, coincident radio and x-ray detections, or bipolar outflow detections. While CO line wings extend to $pm 10-20$ km s$^{-1}$ for the more evolved sources GSS 30 IRS3 and IRS1, CO emission toward other sources, where detected, only extends a few km s$^{-1}$ from the cloud $v_mathrm{LSR}$. The dust spectral index toward the compact objects suggests that the disks are either optically thick at 1.1 mm, or that significant grain growth has already occurred. We identify, for the first time, a single compact continuum source ($R sim 100$ au) toward N6 embedded within a larger continuum structure. SM1N is extended in the continuum but is highly centrally concentrated, with a density profile that follows a $r^{-1.3}$ power law within 200 au, and additional structure suggested by the uv-data. Both N6 and SM1N show no clear bipolar outflows with velocities greater than a few km s$^{-1}$ from the cloud velocity. These sources are candidates to be the youngest protostars or first hydrostatic cores in the Ophiuchus molecular cloud.
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