The emergence of multi-parametric magnetic resonance imaging (mpMRI) has had a profound impact on the diagnosis of prostate cancers (PCa), which is the most prevalent malignancy in males in the western world, enabling a better selection of patients for confirmation biopsy. However, analyzing these images is complex even for experts, hence opening an opportunity for computer-aided diagnosis systems to seize. This paper proposes a fully automatic system based on Deep Learning that takes a prostate mpMRI from a PCa-suspect patient and, by leveraging the Retina U-Net detection framework, locates PCa lesions, segments them, and predicts their most likely Gleason grade group (GGG). It uses 490 mpMRIs for training/validation, and 75 patients for testing from two different datasets: ProstateX and IVO (Valencia Oncology Institute Foundation). In the test set, it achieves an excellent lesion-level AUC/sensitivity/specificity for the GGG$\geq$2 significance criterion of 0.96/1.00/0.79 for the ProstateX dataset, and 0.95/1.00/0.80 for the IVO dataset. Evaluated at a patient level, the results are 0.87/1.00/0.375 in ProstateX, and 0.91/1.00/0.762 in IVO. Furthermore, on the online ProstateX grand challenge, the model obtained an AUC of 0.85 (0.87 when trained only on the ProstateX data, tying up with the original winner of the challenge). For expert comparison, IVO radiologist's PI-RADS 4 sensitivity/specificity were 0.88/0.56 at a lesion level, and 0.85/0.58 at a patient level. Additional subsystems for automatic prostate zonal segmentation and mpMRI non-rigid sequence registration were also employed to produce the final fully automated system. The code for the ProstateX-trained system has been made openly available at https://github.com/OscarPellicer/prostate_lesion_detection. We hope that this will represent a landmark for future research to use, compare and improve upon.