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Breast cancer is a common malignancy and a leading cause of cancer-related deaths in women worldwide. Its early diagnosis can significantly reduce the morbidity and mortality rates in women. To this end, histopathological diagnosis is usually followed as the gold standard approach. However, this process is tedious, labor-intensive, and may be subject to inter-reader variability. Accordingly, an automatic diagnostic system can assist to improve the quality of diagnosis. This paper presents a deep learning approach to automatically classify hematoxylin-eosin-stained breast cancer microscopy images into normal tissue, benign lesion, in situ carcinoma, and invasive carcinoma using our collected dataset. Our proposed model exploited six intermediate layers of the Xception (Extreme Inception) network to retrieve robust and abstract features from input images. First, we optimized the proposed model on the original (unnormalized) dataset using 5-fold cross-validation. Then, we investigated its performance on four normalized datasets resulting from Reinhard, Ruifrok, Macenko, and Vahadane stain normalization. For original images, our proposed framework yielded an accuracy of 98% along with a kappa score of 0.969. Also, it achieved an average AUC-ROC score of 0.998 as well as a mean AUC-PR value of 0.995. Specifically, for in situ carcinoma and invasive carcinoma, it offered sensitivity of 96% and 99%, respectively. For normalized images, the proposed architecture performed better for Makenko normalization compared to the other three techniques. In this case, the proposed model achieved an accuracy of 97.79% together with a kappa score of 0.965. Also, it attained an average AUC-ROC score of 0.997 and a mean AUC-PR value of 0.991. Especially, for in situ carcinoma and invasive carcinoma, it offered sensitivity of 96% and 99%, respectively. These results demonstrate that our proposed model outperformed the baseline AlexNet as well as state-of-the-art VGG16, VGG19, Inception-v3, and Xception models with their default settings. Furthermore, it can be inferred that although stain normalization techniques offered competitive performance, they could not surpass the results of the original dataset.