We present the results of a high angular resolution ( 0 \farcs 3 \approx 40 AU ) Submillimeter Array survey of the 345 GHz ( 870 \mu m ) thermal continuum emission from 9 of the brightest , and therefore most massive , circumstellar disks in the \sim 1 Myr-old Ophiuchus star-forming region . Using two-dimensional radiative transfer calculations , we simultaneously fit the observed continuum visibilities and broadband spectral energy distribution for each disk with a parametric structure model . Compared to previous millimeter studies , this survey includes significant upgrades in modeling , data quality , and angular resolution that provide improved constraints on key structure parameters , particularly those that characterize the spatial distribution of mass in the disks . In the context of a surface density profile motivated by similarity solutions for viscous accretion disks , \Sigma \propto ( R / R _ { c } ) ^ { - \gamma } \exp { [ - ( R / R _ { c } ) ^ { 2 - \gamma } ] } , the best-fit models for the sample disks have characteristic radii R _ { c } \approx 20 -200 AU , high disk masses M _ { d } \approx 0.005 -0.14 M _ { \odot } ( a sample selection bias ) , and a narrow range of radial \Sigma gradients ( \gamma \approx 0.4 -1.0 ) around a median \gamma = 0.9 . These density structures are used in conjunction with accretion rate estimates from the literature to help characterize the viscous evolution of the disk material . Using the standard prescription for disk viscosities , those combined constraints indicate that \alpha \approx 0.0005 -0.08 . Three of the sample disks show large ( R \approx 20 -40 AU ) central cavities in their continuum emission morphologies , marking extensive zones where dust has been physically removed and/or has significantly diminished opacities . Based on the current requirements of planet formation models , these emission cavities and the structure constraints for the sample as a whole suggest that these young disks may eventually produce planetary systems , and have perhaps already started .