Measurement of head biometrics from fetal ultrasonography images is of key importance in monitoring the healthy development of fetuses. However, the accurate measurement of relevant anatomical structures is subject to large inter-observer variability in the clinic. To address this issue, an automated method utilizing Fully Convolutional Networks (FCN) is proposed to determine measurements of fetal head circumference (HC) and biparietal diameter (BPD). An FCN was trained on approximately 2000 2D ultrasound images of the head with annotations provided by 45 different sonographers during routine screening examinations to perform semantic segmentation of the head. An ellipse is fitted to the resulting segmentation contours to mimic the annotation typically produced by a sonographer. The models performance was compared with inter-observer variability, where two experts manually annotated 100 test images. Mean absolute model-expert error was slightly better than inter-observer error for HC (1.99mm vs 2.16mm), and comparable for BPD (0.61mm vs 0.59mm), as well as Dice coefficient (0.980 vs 0.980). Our results demonstrate that the model performs at a level similar to a human expert, and learns to produce accurate predictions from a large dataset annotated by many sonographers. Additionally, measurements are generated in near real-time at 15fps on a GPU, which could speed up clinical workflow for both skilled and trainee sonographers.