Given the outbreak of COVID-19 pandemic and the shortage of medical resource, extensive deep learning models have been proposed for automatic COVID-19 diagnosis, based on 3D computed tomography (CT) scans. However, the existing models independently process the 3D lesion segmentation and disease classification, ignoring the inherent correlation between these two tasks. In this paper, we propose a joint deep learning model of 3D lesion segmentation and classification for diagnosing COVID-19, called DeepSC-COVID, as the first attempt in this direction. Specifically, we establish a large-scale CT database containing 1,805 3D CT scans with fine-grained lesion annotations, and reveal 4 findings about lesion difference between COVID-19 and community acquired pneumonia (CAP). Inspired by our findings, DeepSC-COVID is designed with 3 subnets: a cross-task feature subnet for feature extraction, a 3D lesion subnet for lesion segmentation, and a classification subnet for disease diagnosis. Besides, the task-aware loss is proposed for learning the task interaction across the 3D lesion and classification subnets. Different from all existing models for COVID-19 diagnosis, our model is interpretable with fine-grained 3D lesion distribution. Finally, extensive experimental results show that the joint learning framework in our model significantly improves the performance of 3D lesion segmentation and disease classification in both efficiency and efficacy.