We report the measurement of the elongational stress growth coefficient of a dilute solution of lambda-phage DNA molecules in sugar and corn-syrup solvent mixtures under excess salt conditions. The stress growth is measured using a filament stretching rheometer, at strain rates far exceeding the inverse relaxation time of the DNA molecule. The DNA molecule is assumed to behave like a long chain polymer molecule under good solvent conditions, and the observed behavior is modeled using a bead-spring chain model for the DNA. Excluded-volume (EV) effects are accounted for by a narrow-Gaussian repulsive potential between the beads, hydrodynamic interactions (111) between beads are approximated by the Rotne-Prager-Yamakawa tensor, and the finite length of the DNA molecule is treated by springs whose force-extension behavior mimics that of a wormlike chain. By combining a Brownian dynamics simulation of the chain along with the recently introduced method of successive fine graining [Sunthar, P.; Prakash, J. R. Macromolecules 2005, 38, 6171, predictions of the stress growth coefficient are obtained which are insensitive to the microscopic interaction parameters for EV and HI.