A dilute solution of flexible macromolecules is modeled as a dilute suspension of dumbbells with slightly nonlinear elastic connectors (nearly‐Hookean dumbbells) and subject to Browian motion. The molecular variables are eliminated directly from the kinetic theory by means of a perturbation scheme to give an expression for the stress tensor which is good for flows that do not deform the dumbbells drastically. The stress tensor is expressed in terms of several structure tensors, each of which is given by an auxiliary differential equation. This Hand form of the rheological equation of state can also be expressed in an Oldroyd form or as a memory integral expansion. Calculations of rheological properties for the model show substantial improvements over Hookean dumbbell predictions. In an appendix the distribution function for the nearly‐Hookean dumbbell is derived, and it can be used to obtain the same rheological equation of state by traditional methods.