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Predicting fetal weight at birth is an important aspect of perinatal care, particularly in the context of antenatal management, which includes the planned timing and the mode of delivery. Accurate prediction of weight using prenatal ultrasound is challenging as it requires images of specific fetal body parts during advanced pregnancy which is difficult to capture due to poor quality of images caused by the lack of amniotic fluid. As a consequence, predictions which rely on standard methods often suffer from significant errors. In this paper we propose the Residual Transformer Module which extends a 3D ResNet-based network for analysis of 2D+t spatio-temporal ultrasound video scans. Our end-to-end method, called BabyNet, automatically predicts fetal birth weight based on fetal ultrasound video scans. We evaluate BabyNet using a dedicated clinical set comprising 225 2D fetal ultrasound videos of pregnancies from 75 patients performed one day prior to delivery. Experimental results show that BabyNet outperforms several state-of-the-art methods and estimates the weight at birth with accuracy comparable to human experts. Furthermore, combining estimates provided by human experts with those computed by BabyNet yields the best results, outperforming either of other methods by a significant margin. The source code of BabyNet is available at https://github.com/SanoScience/BabyNet.