In a previous paper, counterion condensation theory was extended by invoking a new alternative auxiliary assumption in order to treat isolated polyions with alternative geometries (beyond a line charge) at finite salt concentrations. Now this theory is further extended to treat a cell model and the osmotic pressures of relatively concentrated DNA solutions in equilibrium with bathing solutions of different salt concentration. Results of the present theory are compared with recently published experimental data of Raspaud et al. and with theoretical calculations of Hansen et al., who combined zero-salt solutions of the nonlinear Poisson-Boltzmann equation for a uniformly charged cylinder with the Donnan approximation. Surprisingly good agreement with the experimental data is observed, especially at the higher DNA concentrations. Agreement with the calculations of Hansen et al. is also rather good at the higher DNA concentrations, but at lower DNA concentrations the results of Hansen et al. fall significantly below those of the present theory. This discrepancy is due in part to the fact that polyion screening by the invading salt is omitted in the theory of Hansen et al. The level of overall agreement between the predictions of this extended counterion condensation cell-model theory and either the experimental data or the calculations of Hansen et al. supports the alternative auxiliary assumption that underlies the present theory.