We have re-analyzed Solar Ultraviolet Measurement of Emitted Radiation (SUMER) observations of a parcel of coronal gas using new collisional ionization equilibrium (CIE) calculations. These improved CIE fractional abundances were calculated using state-of-the-art electron-ion recombination data for K-shell, L-shell, Na-like, and Mg-like ions of all elements from H through Zn and, additionally, Al- through Ar-like ions of Fe. They also incorporate the latest recommended electron impact ionization data for all ions of H through Zn. Improved CIE calculations based on these recombination and ionization data are presented here. We have also developed a new systematic method for determining the average emission measure (EM) and electron temperature (T_e ) of an isothermal plasma. With our new CIE data and a new approach for determining average EM and T_e , we have re-analyzed SUMER observations of the solar corona. We have compared our results with those of previous studies and found some significant differences for the derived EM and T_e . We have also calculated the enhancement of coronal elemental abundances compared to their photospheric abundances, using the SUMER observations themselves to determine the abundance enhancement factor for each of the emitting elements. Our observationally derived first ionization potential factors are in reasonable agreement with the theoretical model of Laming.