This work provides necessary conditions for optimality in problems of optimal control expressed as instances of the generalized problem of Bolza, with the added feature that the fundamental planning interval is allowed to vary. A central product of the analysis is a generalization of the conservation-of-Hamiltonian condition for problems on either fixed or variable intervals. The results, which allow for unprecedented generality in the problem data, are derived from known properties of fixed-interval problems under the hypothesis that the time-dependence of the objective integrand has the same modest level of regularity as the state-dependence.