Deep convolutional neural networks (CNNs) have emerged as a new paradigm for Mammogram diagnosis. Contemporary CNN-based computer-aided-diagnosis systems (CADs) for breast cancer directly extract latent features from input mammogram image and ignore the importance of morphological features. In this paper, we introduce a novel end-to-end deep learning framework for mammogram image processing, which computes mass segmentation and simultaneously predicts diagnosis results. Specifically, our method is constructed in a dual-path architecture that solves the mapping in a dual-problem manner, with an additional consideration of important shape and boundary knowledge. One path, called the Locality Preserving Learner (LPL), is devoted to hierarchically extracting and exploiting intrinsic features of the input. Whereas the other path, called the Conditional Graph Learner (CGL), focuses on generating geometrical features via modeling pixel-wise image to mask correlations. By integrating the two learners, both the cancer semantics and cancer representations are well learned, and the component learning paths in return complement each other, contributing an improvement to the mass segmentation and cancer classification problem at the same time. In addition, by integrating an automatic detection set-up, the DualCoreNet achieves fully automatic breast cancer diagnosis practically. Experimental results show that in benchmark DDSM dataset, DualCoreNet has outperformed other related works in both segmentation and classification tasks, achieving 92.27% DI coefficient and 0.85 AUC score. In another benchmark INbreast dataset, DualCoreNet achieves the best mammography segmentation (93.69% DI coefficient) and competitive classification performance (0.93 AUC score).