The development of machine learning systems for the diagnosis of rare diseases is challenging, mainly due to the lack of data to study them. This paper surmounts this obstacle and presents the first Computer-Aided Diagnosis (CAD) system for low-prevalence collagen VI-related congenital muscular dystrophies. The proposed CAD system works on images of fibroblast cultures obtained with a confocal microscope and relies on a Convolutional Neural Network (CNN) to classify patches of such images in two classes: samples from healthy persons and samples from persons affected by a collagen VI-related muscular distrophy. This fine-grained classification is then used to generate an overall diagnosis on the query image using a majority voting scheme. The proposed system is advantageous, as it overcomes the lack of training data, points to the possibly problematic areas in the query images, and provides a global quantitative evaluation of the condition of the patients, which is fundamental to monitor the effectiveness of potential therapies. The system achieves a high classification performance, with 95% of accuracy and 92% of precision on randomly selected independent test images, outperforming alternative approaches by a significant margin.