Quantum communications are gaining momentum in finding applications in a wide range of domains, especially those require high-security data transmissions. On the other hand, machine learning has achieved numerous breakthrough successes in various application domains including networking. However, currently, machine learning is not as much utilized in quantum networking as in other areas. With such motivation, we propose a machine-learning-powered entanglement routing scheme for quantum networks that aims to accommodate maximum numbers of demands (source-destination pairs) within a time window. More specifically, we present a deep reinforcement routing scheme that is called Deep Quantum Routing Agent (DQRA). In short, DQRA utilizes an empirically designed deep neural network that observes the current network states to schedule the network's demands which are then routed by a qubit-preserved shortest path algorithm. DQRA is trained towards the goal of maximizing the number of resolved requests in each routing window by using an explicitly designed reward function. Our experiment study shows that, on averse, DQRA is able to maintain a rate of successfully routed requests above 80% in a qubit-limited grid network, and about 60% in extreme conditions i.e. each node can act as a repeater exactly once within a window. Furthermore, we show that the complexity and the computational time of DQRA are polynomial in terms of the sizes of the quantum networks.