We present the first central velocity dispersions (sigma_o) measured from the 0.85 micron Calcium II Triplet (CaT) for 8 advanced (i.e. single nuclei) local (z < 0.15) Ultraluminous Infrared Galaxies (ULIRGs). First, these measurements are used to test the prediction that the sigma-Discrepancy, in which the CaT sigma_o is systematically larger than the sigma_o obtained from the 1.6 or 2.3 micron stellar CO band-heads, extends to ULIRG luminosities. Next, we combine the CaT data with rest-frame I-band photometry obtained from archival Hubble Space Telescope data and the Sloan Digital Sky Survey (SDSS) to derive dynamical properties for the 8 ULIRGs. These are then compared to the dynamical properties of 9,255 elliptical galaxies from the SDSS within the same redshift volume and of a relatively nearby (z < 0.4) sample of 53 QSO host galaxies. A comparison is also made between the I-band and H-band dynamical properties of the ULIRGs. We find four key results: 1) the sigma-Discrepancy extends to ULIRG luminosities; 2) at I-band ULIRGs lie on the Fundamental Plane (FP) in a region consistent with the most massive elliptical galaxies and not low-intermediate mass ellipticals as previously reported in the near-infrared; 3) the I-band M/L of ULIRGs are consistent with an old stellar population, while at H-band ULIRGs appear significantly younger and less massive; and 4) we derive an I-band Kormendy Relation from the SDSS ellipticals and demonstrate that ULIRGs and QSO host galaxies are dynamically similar.