We describe a model for the speed of an internal bore as a function of amplitude in continuous stratification of arbitrary form. The model is developed from the Dubreil-Jacotin–Long theory for nonlinear solitary waves in the conjugate flow limit, which represents an internal hydraulic jump, by allowing dissipation across the jump. The bore speeds predicted by the model are consistent in both the small- and large-amplitude limits with the waveguide intrinsic to the ambient stratification. The model therefore represents a significant advancement over previous theories limited to sharp two-layer stratification. The model shows good agreement with Navier–Stokes simulations of both undular and turbulent internal bores generated by dam break into a continuously stratified ambient with a finite pycnocline, predicting both the front speed as well as the velocity and density structure through the bore. A model is required for the structure of the energy dissipation, and we introduce a one-parameter closure that produces excellent agreement with numerical results, particularly in the parameter limit that maximizes the overall dissipation. By varying the dissipation parameter, the model reproduces previous two-layer theories in the thin-pycnocline limit, and suggests an improved two-layer front speed relationship. It is demonstrated that, even for the sharp two-layer limit, continuous stratification, and particularly the nonlinear waveguide, must be accounted for in order to accurately predict the bore speed and structure.