The number of mitotic cells present in histopathological slides is an important predictor of tumor proliferation in the diagnosis of breast cancer. However, the current approaches can hardly perform precise pixel-level prediction for mitosis datasets with only weak labels (i.e., only provide the centroid location of mitotic cells), and take no account of the large domain gap across histopathological slides from different pathology laboratories. In this work, we propose a Domain adaptive Box-supervised Instance segmentation Network (DBIN) to address the above issues. In DBIN, we propose a high-performance Box-supervised Instance-Aware (BIA) head with the core idea of redesigning three box-supervised mask loss terms. Furthermore, we add a Pseudo-Mask-supervised Semantic (PMS) head for enriching characteristics extracted from underlying feature maps. Besides, we align the pixel-level feature distributions between source and target domains by a Cross-Domain Adaptive Module (CDAM), so as to adapt the detector learned from one lab can work well on unlabeled data from another lab. The proposed method achieves state-of-the-art performance across four mainstream datasets. A series of analysis and experiments show that our proposed BIA and PMS head can accomplish mitosis pixel-wise localization under weak supervision, and we can boost the generalization ability of our model by CDAM.