The composition of ultra-high-energy cosmic rays is measured with the High Resolution Fly's Eye cosmic-ray observatory data using the X sub(max) technique. Data were collected in stereo between 1999 November and 2001 September. The data are reconstructed with well-determined geometry. Measurements of the atmospheric transmission are incorporated in the reconstruction. The detector resolution is found to be 30 g cm super(-2) in X sub(max) and 13% in energy. The X sub(max) elongation rate between 10 super(18.0) and 10 super(19.4) eV is measured to be 54.5 c 6.5 (stat) c 4.5 (sys) g cm super(-2) per decade. This is compared with predictions using the QGSJet01 and SIBYLL 2.1 hadronic interaction models for both protons and iron nuclei. CORSIKA-generated extensive air showers are incorporated directly into a detailed detector Monte Carlo program. The elongation rate and the X sub(max) distribution widths are consistent with a constant or slowly changing and predominantly light composition. A simple model containing only protons and iron nuclei is compared with QGSJet and SIBYLL. The best agreement between the model and the data is for 80% protons for QGSJet and 60% protons for SIBYLL.