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We consider a class of nonlinear differential equations that arises in the study of chemical reaction systems known to be locally asymptotically stable and prove that they are in fact globally asymptotically stable. More specifically, we will consider chemical reaction systems that are weakly reversible, have a deficiency of zero, and are equipped with mass action kinetics. We show that if for each $c \in \mathbb{R}_{>0}^m$ the intersection of the stoichiometric compatibility class $c + S$ with the subsets on the boundary that could potentially contain equilibria, $L_W$, are at most discrete, then global asymptotic stability follows. Previous global stability results for the systems considered in this paper required $(c + S) \cap L_W = \emptyset$ for each $c \in \mathbb{R}^m_{> 0}$.