Currently, autism spectrum disorder (ASD) is mainly diagnosed by the observation of core behavioral symptoms. Consequently, the window of opportunity for effective intervention may have passed, when the disorder is detected until 3 years of age. Thus, it is of great importance to identify imaging-based biomarkers for early diagnosis of ASD. Previous findings indicate that an abnormal pattern of the amygdala and hippocampal development in autism persists through childhood and adolescence. However, due to the low tissue contrast and small structural size of amygdala and hippocampal subfields, our knowledge on their growth in autistics in early stage still remains very limited. In this paper, for the first time, we propose a volume-based analysis of the amygdala and hippocampal subfields of the infant subjects with risk of ASD at around 24 months of age. Specifically, to address the challenge of low tissue contrast, we propose a novel deep-learning approach, i.e., dilated-dense U-Net, to automatically segment the amygdala and hippocampal subfields. Experimental results on National Database for Autism Research (NDAR) show the advantages of our proposed method in terms of segmentation accuracy. Our volume-based analysis shows the overgrowths of amygdala and CA1-3 of hippocampus, which may link to the emergence of autism spectrum disorder.