The goal of binary classification is to identify whether an input sample belongs to positive or negative classes. Usually, supervised learning is applied to obtain a classification rule, but in real-world applications, it is conceivable that only positive and unlabeled data are accessible for learning, which is called learning from positive and unlabeled data (PU learning). Furthermore, in practice, data distributions are likely to differ between training and testing due to, for example, time variation and domain shift. The covariate shift is a dataset shift situation, where distributions of covariates (inputs) differ between training and testing, but the input-output relation is the same. In this paper, we address the PU learning problem under the covariate shift. We propose an importanceweighted PU learning method and reveal in which situations the importance-weighting is necessary. Moreover, we derive the convergence rate of the proposed method under mild conditions and experimentally demonstrate its effectiveness.