A new method for estimating tropical cyclone track uncertainty is presented and tested. This method uses a neural network to predict a bivariate normal distribution, which serves as an estimate for track uncertainty. We train the network and make predictions on forecasts from the National Hurricane Center (NHC), which currently uses static error distributions based on forecasts from the past five years for most applications. The neural network-based method produces uncertainty estimates that are dynamic and probabilistic. Further, the neural network-based method allows for probabilistic statements about tropical cyclone trajectories, including landfall probability, which we highlight. We show that our predictions are well calibrated using multiple metrics, that our method produces better uncertainty estimates than current NHC approaches, and that our method achieves similar performance to the Global Ensemble Forecast System. Once trained, the computational cost of predictions using this method is negligible, making it a strong candidate to improve the NHC's operational estimations of tropical cyclone track uncertainty.