Coronal mass ejections detected with the Solar Maximum Mission coronagraph/polarimeter are often accompanied by erupting prominence material observed both in H&agr; and in the electron scattering continuum. H&agr; emission is concentrated in bright filaments moving radially outward. The same filaments are seen in the electron scattering continuum as regions of enhanced brightness. In this paper we develop a diagnostic method based on the observed H&agr; and continuum brightness to derive the electron density, line of sight thickness, and degree of ionization of hydrogen as functions of the temperature of the prominence filaments. Our method differs from that of Poland and Munro (1976) in the treatment of Ly &agr; excitation. Analysis of data from the event of August 18, 1980, illustrates that the rising prominence material has decreased density, increased temperature, and increased ionization of hydrogen relative to quiescent prominences in the lower corona. Hydrogen is found to be 90--99% ionized, electron densities are near 108 cm-3, and the temperature is near 20,000 K. The increased ionization is due mainly to the decreased density. Use of the results is made here and in an accompanying paper by Illing and Hundhausen to determine the total mass ejected in the event.