We present a high angular resolution ($sim 0.2^{primeprime}$), high sensitivity ($sigma sim 0.2$ mJy) survey of the 870 $mu$m continuum emission from the circumstellar material around 49 pre-main sequence stars in the $rho$ Ophiuchus molecular cloud. Because most millimeter instruments have resided in the northern hemisphere, this represents the largest high-resolution, millimeter-wave survey of the circumstellar disk content of this cloud. Our survey of 49 systems comprises 63 stars; we detect disks associated with 29 single sources, 11 binaries, 3 triple systems and 4 transition disks. We present flux and radius distributions for these systems; in particular, this is the first presentation of a reasonably complete probability distribution of disk radii at millimeter-wavelengths. We also compare the flux distribution of these protoplanetary disks with that of the disk population of the Taurus-Auriga molecular cloud. We find that disks in binaries are both significantly smaller and have much less flux than their counterparts around isolated stars. We compute truncation calculations on our binary sources and find that these disks are too small to have been affected by tidal truncation and posit some explanations for this. Lastly, our survey found 3 candidate gapped disks, one of which is a newly identified transition disk with no signature of a dip in infrared excess in extant observations.
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