We present the results of optically thick non-LTE radiative transfer calculations of lines and continua of H, C I-IV, and O I-VI and other elements using a new one-dimensional, time-independent model corresponding to the average quiet-Sun chromosphere and transition region. The model is based principally on the Curdt et al. SUMER atlas of the extreme ultraviolet spectrum. Our model of the chromosphere is a semiempirical one, with the temperature distribution adjusted to obtain optimum agreement between calculated and observed continuum intensities, line intensities, and line profiles. Our model of the transition region is determined theoretically from a balance between (a) radiative losses and (b) the downward energy flow from the corona due to thermal conduction and particle diffusion, and using boundary conditions at the base of the transition region established at the top of the chromosphere from the semiempirical model. The quiet-Sun model presented here should be considered as a replacement of the earlier model C of Vernazza et al., since our new model is based on an energy-balance transition region, a better underlying photospheric model, a more extensive set of chromospheric observations, and improved calculations. The photospheric structure of the model given here is the same as in Table 3 of Fontenla, Avrett, Thuiller, & Harder. We show comparisons between calculated and observed continua, and between the calculated and observed profiles of all significant lines of H, C I-IV, and O I-VI in the wavelength range 67-173 nm. While some of the calculated lines are not in emission as observed, we find reasonable general agreement, given the uncertainties in atomic rates and cross sections, and we document the sources of the rates and cross sections used in the calculation. We anticipate that future improvements in the atomic data will give improved agreement with the observations.