Semantic communications have been envisioned as a potential technique that goes beyond Shannon paradigm. Unlike modern communications that provide bit-level security, the eaves-dropping of semantic communications poses a significant risk of potentially exposing intention of legitimate user. To address this challenge, a novel deep neural network (DNN) enabled secure semantic communication (DeepSSC) system is developed by capitalizing on physical layer security. To balance the tradeoff between security and reliability, a two-phase training method for DNNs is devised. Particularly, Phase I aims at semantic recovery of legitimate user, while Phase II attempts to minimize the leakage of semantic information to eavesdroppers. The loss functions of DeepSSC in Phases I and II are respectively designed according to Shannon capacity and secure channel capacity, which are approximated with variational inference. Moreover, we define the metric of secure bilingual evaluation understudy (S-BLEU) to assess the security of semantic communications. Finally, simulation results demonstrate that DeepSSC achieves a significant boost to semantic security particularly in high signal-to-noise ratio regime, despite a minor degradation of reliability.