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The properties of Galactic molecular clouds tabulated by Solomon etal (1987) (SRBY) are re-examined using the Boston University-FCRAO Galactic Ring Survey of 13CO J=1-0 emission. These new data provide a lower opacity tracer of molecular clouds and improved angular and spectral resolution than previous surveys of molecular line emission along the Galactic Plane. We calculate GMC masses within the SRBY cloud boundaries assuming LTE conditions throughout the cloud and a constant H2 to 13CO abundance, while accounting for the variation of the 12C/13C with Galacto-centric radius. The LTE derived masses are typically five times smaller than the SRBY virial masses. The corresponding median mass surface density of molecular hydrogen for this sample is 42 Msun/pc^2, which is significantly lower than the value derived by SRBY (median 206 Msun/pc^2) that has been widely adopted by most models of cloud evolution and star formation. This discrepancy arises from both the extrapolation by SRBY of velocity dispersion, size, and CO luminosity to the 1K antenna temperature isophote that likely overestimates the GMC masses and our assumption of constant 13CO abundance over the projected area of each cloud. Owing to the uncertainty of molecular abundances in the envelopes of clouds, the mass surface density of giant molecular clouds could be larger than the values derived from our 13CO measurements. From velocity dispersions derived from the 13CO data, we find that the coefficient of the cloud structure functions, vo=sigma_v/R^{1/2}, is not constant, as required to satisfy Larson's scaling relationships, but rather systematically varies with the surface density of the cloud as Sigma^{0.5} as expected for clouds in self-gravitational equlibrium., Accepted by ApJ. Newest version includes modifications from the referee