The equilibrium distribution of dilute solutes between vapor and liquid coexisting phases of near-critical solvents exhibits a simple dependence on the solvent's liquid density which extends over a wide temperature range; however, theory predicts this dependence only as an asymptotic limit. In order to test quantitatively the extension of this behavior and compare it with the value predicted from the asymptotic relationship, a novel high-pressure apparatus with a sampling manifold, which avoids perturbing the system during sampling, was used to measure the equilibrium concentrations of azulene in coexisting liquid and vapor carbon dioxide contained in a high pressure sapphire cell. The density and composition of the two phases under equilibrium were determined between 283 and 301 K. The linearity of RT ln K-D(K-D being the distribution equilibrium constant) in the solvent's liquid density was verified over the entire experimental temperature range, and the value of the slope was close to that corresponding to the asymptotic description as calculated with the hydrostatic hypernetted chain theory. An explanation for this observation lies in the weak temperature dependence of the Krichevskii function for this system; however a full quantitative description of the observed phenomenon remains a challenge, which should be provided by crossover theory.