The Lincoln-Petersen estimator is one of the most popular estimators used in capture-recapture studies. It was developed for a sampling situation in which two sources independently identify members of a target population. For each of the two sources, it is determined if a unit of the target population is identified or not. This leads to a 2 x 2 table with frequencies f(11), f(10), f(01), f(00) indicating the number of units identified by both sources, by the first but not the second source, by the second but not the first source and not identified by any of the two sources, respectively. However, f(00) is unobserved so that the 2 x 2 table is incomplete and the Lincoln-Petersen estimator provides an estimate for f(00). In this paper, we consider a generalization of this situation for which one source provides not only a binary identification outcome but also a count outcome of how many times a unit has been identified. Using a truncated Poisson count model, truncating multiple identifications larger than two, we propose a maximum likelihood estimator of the Poisson parameter and, ultimately, of the population size. This estimator shows benefits, in comparison with Lincoln-Petersen's, in terms of bias and efficiency. It is possible to test the homogeneity assumption that is not testable in the Lincoln-Petersen framework. The approach is applied to surveillance data on syphilis from Izmir, Turkey. Copyright (c) 2014 John Wiley & Sons, Ltd.