  Motivated by the limitations of the traditional definitions of the
Riemann-Stieltjes and Darboux-Stieltjes integrals, we introduce a generalized
Darboux-Stieltjes integral that is equivalent to an earlier generalization by
Ross \cite{Ross}. Our definition builds upon an approach to the
Darboux-Stieltjes integral recently introduced by the first author and
Convertito \cite{TSI}. We show that our definition agrees with all previous
definitions, but that the class of integrable functions is much larger. We
develop all the analogs of the classic results for the Riemann integral, and
rectify the problems inherent in the definition of the Darboux-Stieltjes
integral in \cite{TSI}. In particular, we show the Bounded Convergence Theorem
holds for our definition and that it agrees with the Lebesgue-Stieltjes
integral.
