A baby's gestational age determines whether or not they are premature, which helps clinicians decide on suitable post-natal treatment. The most accurate dating methods use Ultrasound Scan (USS) machines, but these are expensive, require trained personnel and cannot always be deployed to remote areas. In the absence of USS, the Ballard Score, a postnatal clinical examination, can be used. However, this method is highly subjective and results vary widely depending on the experience of the examiner. Our main contribution is a novel system for automatic postnatal gestational age estimation using small sets of images of a newborn's face, foot and ear. Our two-stage architecture makes the most out of Convolutional Neural Networks trained on small sets of images to predict broad classes of gestational age, and then fuses the outputs of these discrete classes with a baby's weight to make fine-grained predictions of gestational age using Support Vector Regression. On a purpose-collected dataset of 130 babies, experiments show that our approach surpasses current automatic state-of-the-art postnatal methods and attains an expected error of 6 days. It is three times more accurate than the Ballard method. Making use of images improves predictions by 33% compared to using weight only. This indicates that even with a very small set of data, our method is a viable candidate for postnatal gestational age estimation in areas were USS is not available.