This paper describes a low-cost, robust, and accurate remote eye-tracking system that uses an industrial prototype smartphone with integrated infrared illumination and camera. Numerous studies have demonstrated the beneficial use of eye-tracking in domains such as neurological and neuropsychiatric testing, advertising evaluation, pilot training, and automotive safety. Remote eye-tracking on a smartphone could enable the significant growth in the deployment of applications in these domains. Our system uses a 3D gaze-estimation model that enables accurate point-of-gaze (PoG) estimation with free head and device motion. To accurately determine the input eye features (pupil center and corneal reflections), the system uses Convolutional Neural Networks (CNNs) together with a novel center-of-mass output layer. The use of CNNs improves the system’s robustness to the significant variability in the appearance of eye-images found in handheld eye trackers. The system was tested with 8 subjects with the device free to move in their hands and produced a gaze bias of 0.72°. Our hybrid approach that uses artificial illumination, a 3D gaze-estimation model, and a CNN feature extractor achieved an accuracy that is significantly (400%) better than current eye-tracking systems on smartphones that use natural illumination and machine-learning techniques to estimate the PoG.