This paper presents the problem of identifying a one-place unbounded Petri net. Given an unlabelled graph that represents the modified coverability graph of a net, we establish a Petri net model whose modified coverability graph is isomorphic to the unlabelled graph, and that can identify the weight of the arcs that cannot be obtained from the coverability graph of the net. Based on the partition of the nodes in the unlabelled graph, we guess and decide the structure of a Petri net and the weight of the arcs by an integer linear programming problem. The unknowns to be determined are the elements of the pre- and post-incidence matrices and the initial marking of the net, which can be computed by solving an integer linear programming problem. Finally, an example is used to validate the rationality and effectiveness of the proposed approach.