The current approach to fetal anomaly screening is based on biometric measurements derived from individually selected ultrasound images. In this paper, we introduce a paradigm shift that attains human-level performance in biometric measurement by aggregating automatically extracted biometrics from every frame across an entire scan, with no need for operator intervention. We use a neural network to classify each frame of an ultrasound video recording. We then measure fetal biometrics in every frame where appropriate anatomy is visible. We use a Bayesian method to estimate the true value of each biometric from a large number of measurements and probabilistically reject outliers. We performed a retrospective experiment on 1457 recordings (comprising 48 million frames) of 20-week ultrasound scans, estimated fetal biometrics in those scans and compared our estimates to real-time manual measurements. Our method achieves human-level performance in estimating fetal biometrics and estimates well-calibrated credible intervals for the true biometric value.