Objective. This work investigates the use of deep convolutional neural networks (CNN) to automatically perform measurements of fetal body parts, including head circumference, biparietal diameter, abdominal circumference and femur length, and to estimate gestational age and fetal weight using fetal ultrasound videos. Approach. We developed a novel multi-task CNN-based spatio-temporal fetal US feature extraction and standard plane detection algorithm (called FUVAI) and evaluated the method on 50 freehand fetal US video scans. We compared FUVAI fetal biometric measurements with measurements made by five experienced sonographers at two time points separated by at least two weeks. Intra- and inter-observer variabilities were estimated. Main results. We found that automated fetal biometric measurements obtained by FUVAI were comparable to the measurements performed by experienced sonographers The observed differences in measurement values were within the range of inter- and intra-observer variability. Moreover, analysis has shown that these differences were not statistically significant when comparing any individual medical expert to our model. Significance. We argue that FUVAI has the potential to assist sonographers who perform fetal biometric measurements in clinical settings by providing them with suggestions regarding the best measuring frames, along with automated measurements. Moreover, FUVAI is able perform these tasks in just a few seconds, which is a huge difference compared to the average of six minutes taken by sonographers. This is significant, given the shortage of medical experts capable of interpreting fetal ultrasound images in numerous countries.