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In ultrasound shear wave elasticity (SWE) imaging, a number of algorithms exist for estimating the shear wave speed (SWS) from spatiotemporal displacement data. However, no method provides a well-calibrated and practical uncertainty metric, hindering SWE's clinical adoption and utility in downstream decision-making. Here, we designed a deep learning SWS estimator that simultaneously outputs a quantitative and well-calibrated uncertainty value for each estimate. Our deep neural network (DNN) takes as input a single 2D spatiotemporal plane of tracked displacement data and outputs the two parameters $m$ and $σ$ of a log-normal probability distribution. For training and testing, we used in vivo 2D-SWE data of the cervix collected from 30 pregnant subjects, totaling 551 acquisitions and >2 million space-time plots. Points were grouped by uncertainty into bins to assess uncertainty calibration: the predicted uncertainty closely matched the root-mean-square estimation error, with an average absolute percent deviation of 3.84%. We created a leave-one-out ensemble model that estimated uncertainty with better calibration (1.45%) than any individual ensemble member on a held-out patient's data. Lastly, we applied the DNN to an external dataset to evaluate its generalizability. We have made the trained model, SweiNet, openly available to provide the research community with a fast SWS estimator that also outputs a well-calibrated estimate of the predictive uncertainty.