Automated methods for detecting clinically significant (CS) prostate cancer (PCa) in multi-parameter magnetic resonance images (mp-MRI) are of high demand. Existing methods typically employ several separate steps, each of which is optimized individually without considering the error tolerance of other steps. As a result, they could either involve unnecessary computational cost or suffer from errors accumulated over steps. In this paper, we present an automated CS PCa detection system, where all steps are optimized jointly in an end-to-end trainable deep neural network. The proposed neural network consists of concatenated subnets: 1) a novel tissue deformation network (TDN) for automated prostate detection and multimodal registration and 2) a dual-path convolutional neural network (CNN) for CS PCa detection. Three types of loss functions, i.e., classification loss, inconsistency loss, and overlap loss, are employed for optimizing all parameters of the proposed TDN and CNN. In the training phase, the two nets mutually affect each other and effectively guide registration and extraction of representative CS PCa-relevant features to achieve results with sufficient accuracy. The entire network is trained in a weakly supervised manner by providing only image-level annotations (i.e., presence/absence of PCa) without exact priors of lesions' locations. Compared with most existing systems which require supervised labels, e.g., manual delineation of PCa lesions, it is much more convenient for clinical usage. Comprehensive evaluation based on fivefold cross validation using 360 patient data demonstrates that our system achieves a high accuracy for CS PCa detection, i.e., a sensitivity of 0.6374 and 0.8978 at 0.1 and 1 false positives per normal/benign patient.