Deep-learning-as-a-service is a novel and promising computing paradigm aiming at providing machine/deep learning solutions and mechanisms through Cloud-based computing infrastructures. Thanks to its ability to remotely execute and train deep learning models (that typically require high computational loads and memory occupation), such an approach guarantees high performance, scalability, and availability. Unfortunately, such an approach requires to send information to be processed (e.g., signals, images, positions, sounds, videos) to the Cloud, hence having potentially catastrophic-impacts on the privacy of users. This paper introduces a novel distributed architecture for deep-learning-as-a-service that is able to preserve the user sensitive data while providing Cloud-based machine and deep learning services. The proposed architecture, which relies on Homomorphic Encryption that is able to perform operations on encrypted data, has been tailored for Convolutional Neural Networks (CNNs) in the domain of image analysis and implemented through a client-server REST-based approach. Experimental results show the effectiveness of the proposed architecture.