A new approach to the verification of current-state opacity for discrete event systems is proposed in this paper, which is modeled with unbounded Petri nets. The concept of opacity verification is first extended from bounded Petri nets to unbounded Petri nets. In this model, all transitions and partial places are assumed to be unobservable, i.e., only the number of tokens in the observable places can be measured. In this work, a novel basis coverability graph is constructed by using partial markings and quasi-observable transitions. By this graph, this research finds that an unbounded net system is current-state opaque if, for an arbitrary partial marking, there always exists at least one regular marking in the result of current-state estimation with respect to the partial marking not belonging to the given secret. Finally, a sufficient and necessary condition is proposed for the verification of current-state opacity. A manufacturing system example is presented to illustrate that the concept of current-state opacity can be verified for unbounded net systems.