We develop a crossover theory for dilute polymer solutions, analogous to crossover theories for critical phenomena in simple fluids. In this theory, a critical degree of polymerization N* is found, which plays a similar role as the Ginzburg number in second-order phase transitions. To test the predictions of this theory, we perform Monte Carlo simulations of polymer chains composed of rigidly bonded hard spheres of various diameters and chain lengths. Various properties of these chains were analyzed, including the end-to-end distance distribution and mean-square radius of gyration. We find that the approach to the asymptotic scaling regime displays two types of crossover behavior, depending on the value of the model parameter (u) over bar, which is a measure of the strength of the monomer-monomer excluded volume interaction: (i) (u) over bar < 1 and (ii) (u) over bar > 1. In case (i), the systemexhibits crossover from a Gaussian chain to the Kuhnian chain, as the degree of polymerization increases. In case (ii), the system exhibits crossover from the rigid rod to a Kuhnian chain. Our crossover theory is found to work well for polymers with (u) over bar > 1 only near the asymptotic scaling regime. However, for (u) over bar, the theory works well in all regimes. [S0021-9606(99)50505- 9].