  We present new determinations of disk surface density, independent of an
assumed dust opacity, for a sample of 7 bright, diverse protoplanetary disks
using measurements of disk dust lines. We develop a robust method for
determining the location of dust lines by modeling disk interferometric
visibilities at multiple wavelengths. The disks in our sample have newly
derived masses that are 9-27% of their host stellar mass, substantially larger
than the minimum mass solar nebula. All are stable to gravitational collapse
except for one which approaches the limit of Toomre-Q stability. Our mass
estimates are 2-15 times larger than estimates from integrated optically thin
dust emission. We derive depleted dust-to-gas ratios with typical values of
~$10^{-3}$ in the outer disk. Using coagulation models we derive dust surface
density profiles that are consistent with millimeter dust observations. In
these models, the disks formed with an initial dust mass that is a factor of
~10 greater than is presently observed. Of the three disks in our sample with
resolved CO line emission, the masses of HD 163296, AS 209, and TW Hya are
roughly 3, 115, and 40 times more massive than estimates from CO respectively.
This range indicates that CO depletion is not uniform across different disks
and that dust is a more robust tracer of total disk mass. Our method of
determining surface density using dust lines is robust even if particles form
as aggregates and is useful even in the presence of dust substructure caused by
pressure traps. The low Toomre-Q values observed in this sample indicate that
at least some disks do not accrete efficiently.
