In this article, we discuss the problem of testing for a changepoint in the structure of an integer-valued time series. In particular, we consider a test statistic of cumulative sum type for general Poisson autoregressions of order 1. We investigate the asymptotic behaviour of conditional least-squares estimates of the parameters in the presence of a changepoint. Then, we derive the asymptotic distribution of the test statistic under the hypothesis of no change, allowing for the calculation of critical values. We prove consistency of the test, that is, asymptotic power 1, and consistency of the corresponding changepoint estimate. As an application, we have a look at changepoint detection in daily epileptic seizure counts from a clinical study.