Sarcasm employs ambivalence, where one says something positive but actually means negative, and vice versa. The essence of sarcasm, which is also a sufficient and necessary condition, is the conflict between literal and implied sentiments expressed in one sentence. However, it is difficult to recognize such sentiment conflict because the sentiments are mixed or even implicit. As a result, the recognition of sophisticated and obscure sentiment brings in a great challenge to sarcasm detection. In this paper, we propose a Dual-Channel Framework by modeling both literal and implied sentiments separately. Based on this dual-channel framework, we design the Dual-Channel Network (DC-Net) to recognize sentiment conflict. Experiments on political debates (i.e. IAC-V1 and IAC-V2) and Twitter datasets show that our proposed DC-Net achieves state-of-the-art performance on sarcasm recognition. Our code is released to support research.