The human epidermal growth factor receptor 2 (HER2) gene amplification status is a crucial marker for evaluating clinical therapies of breast or gastric cancer. Therefore, the detection of HER2 gene amplification status is highly relevant in histopathological diagnostics. Recently, the application of convolutional neural networks (CNNs) has shown large improvements in the automation of classification and object detection in medical image analysis. Here, we propose a deep learning-based pipeline for the detection, localization and classification of interphase nuclei depending on their HER2 gene amplification state in Fluorescence in situ hybridization (FISH) images. Our pipeline combines two RetinaNet-based object localization networks which are trained (1) to detect and classify interphase nuclei into distinct classes normal, low-grade and high-grade and (2) to detect and classify FISH signals into distinct classes HER2 or centromere of chromosome 17 (CEN17). By independently classifying each nucleus twice, the two-step pipeline provides both robustness and interpretability for the automated detection of the HER2 amplification status. We demonstrate that the accuracy of this deep learning-based pipeline is on par with that of three pathologists. We demonstrate that our pipeline accurately classifies FISH images on a set of 57 validation images containing several hundreds of nuclei. Consequently, high quality FISH images can now be analyzed at once regarding their image-wide HER2 gene amplification status. The automatic pipeline is a first step towards assisting pathologists in evaluating the HER2 status of tumors using FISH images, for analyzing FISH images in retrospective studies, and for optimizing the documentation of each tumor sample by automatically annotating and reporting of the HER2 gene amplification specificities.