Abstract: Document interpretation and dialog understanding are the two major challenges for conversational machine reading. In this work, we propose "Discern", a discourse-aware entailment reasoning network to strengthen the connection and enhance the understanding of both document and dialog. Specifically, we split the document into clause-like elementary discourse units (EDU) using a pre-trained discourse segmentation model, and we train our model in a weakly-supervised manner to predict whether each EDU is entailed by the user feedback in a conversation. Based on the learned EDU and entailment representations, we either reply to the user our final decision "yes/no/irrelevant" of the initial question, or generate a follow-up question to inquiry more information. Our experiments on the ShARC benchmark (blind, held-out test set) show that Discern achieves state-of-the-art results of 78.3% macro-averaged accuracy on decision making and 64.0 BLEU1 on follow-up question generation. Code and models are released at https://github.com/Yifan-Gao/Discern.