Deep learning (DL) has proven highly effective for ultrasound-based computer-aided diagnosis (CAD) of breast cancers. In an automatic CAD system, lesion detection is critical for the following diagnosis. However, existing DL-based methods generally require voluminous manually-annotated region of interest (ROI) labels and class labels to train both the lesion detection and diagnosis models. In clinical practice, the ROI labels, i.e. ground truths, may not always be optimal for the classification task due to individual experience of sonologists, resulting in the issue of coarse annotation to limit the diagnosis performance of a CAD model. To address this issue, a novel Two-Stage Detection and Diagnosis Network (TSDDNet) is proposed based on weakly supervised learning to improve diagnostic accuracy of the ultrasound-based CAD for breast cancers. In particular, all the initial ROI-level labels are considered as coarse annotations before model training. In the first training stage, a candidate selection mechanism is then designed to refine manual ROIs in the fully annotated images and generate accurate pseudo-ROIs for the partially annotated images under the guidance of class labels. The training set is updated with more accurate ROI labels for the second training stage. A fusion network is developed to integrate detection network and classification network into a unified end-to-end framework as the final CAD model in the second training stage. A self-distillation strategy is designed on this model for joint optimization to further improves its diagnosis performance. The proposed TSDDNet is evaluated on three B-mode ultrasound datasets, and the experimental results indicate that it achieves the best performance on both lesion detection and diagnosis tasks, suggesting promising application potential.