In this paper we present a two-step neural network model to separate detections of solar system objects from optical and electronic artifacts in data obtained with the "Asteroid Terrestrial-impact Last Alert System" (ATLAS), a near-Earth asteroid sky survey system [arXiv:1802.00879]. A convolutional neural network [arXiv:1807.10912] is used to classify small "postage-stamp" images of candidate detections of astronomical sources into eight classes, followed by a multi-layered perceptron that provides a probability that a temporal sequence of four candidate detections represents a real astronomical source. The goal of this work is to reduce the time delay between Near-Earth Object (NEO) detections and submission to the Minor Planet Center. Due to the rare and hazardous nature of NEOs [Harris and D'Abramo, 2015], a low false negative rate is a priority for the model. We show that the model reaches 99.6\% accuracy on real asteroids in ATLAS data with a 0.4\% false negative rate. Deployment of this model on ATLAS has reduced the amount of NEO candidates that astronomers must screen by 90%, thereby bringing ATLAS one step closer to full autonomy.