Determining the success of eradication of an invasive species requires a way to decide when its risk of reoccurrence has become acceptably low. In Japan, the area populated by the Argentine ant, Linepithema humile (Mayr), is expanding, and eradication via chemical treatment is ongoing at various locations. One such program in Tokyo was apparently successful, because the ant population decreased to undetectable levels within a short time. However, construction of a population model for management purposes was difficult because the probability of detecting ants decreases rapidly as the population collapses. To predict the time when the ant was eradicated, we developed a multinomial-mixture model for chemical eradication based on monthly trapping data and the history of pesticide applications. We decided when to declare that eradication had been successful by considering both 'eradication' times, which we associated with eradication probabilities of 95% and 99%, and an optimal stopping time based on a 'minimum expected economic cost' that considered the possibility that surveys were stopped too soon. By applying these criteria, we retroactively declared that Argentine ants had been eradicated 38-42 months after the start of treatments (16-17 months after the last sighting).