Eye tracking is a key technology for gaze-based interactions in Extended Reality (XR), but traditional frame-based systems struggle to meet XR's demands for high accuracy, low latency, and power efficiency. Event cameras offer a promising alternative due to their high temporal resolution and low power consumption. In this paper, we present FACET (Fast and Accurate Event-based Eye Tracking), an end-to-end neural network that directly outputs pupil ellipse parameters from event data, optimized for real-time XR applications. The ellipse output can be directly used in subsequent ellipse-based pupil trackers. We enhance the EV-Eye dataset by expanding annotated data and converting original mask labels to ellipse-based annotations to train the model. Besides, a novel trigonometric loss is adopted to address angle discontinuities and a fast causal event volume event representation method is put forward. On the enhanced EV-Eye test set, FACET achieves an average pupil center error of 0.20 pixels and an inference time of 0.53 ms, reducing pixel error and inference time by 1.6 × and 1.8 × compared to the prior art, EV-Eye, with 4.4 × and 11.7 × less parameters and arithmetic operations. The code is available at https://github.com/DeanJY/FACET.