The efficient integration of satellite and terrestrial networks has become an import-ant component for 6G wireless architectures to provide highly reliable and secure connectivity over a wide geographical area. As the satellite and cellular networks are developed separately these years, the integrated network should syn-ergize the communication, storage, computa-tion capabilities of both sides towards an intel-ligent system more than mere consideration of coexistence. This has motivated us to develop double-edge intelligent integrated satellite and terrestrial networks (DILIGENT). Leveraging the boost development of multi-access edge computing (MEC) technology and artificial in-telligence (AI), the framework is entitled with the systematic learning and adaptive network management of satellite and cellular networks. In this article, we provide a brief review of the state-of-art contributions from the perspective of academic research and standardization. Then we present the overall design of the pro-posed DILIGENT architecture, where the ad-vantages are discussed and summarized. Strat-egies of task offloading, content caching and distribution are presented. Numerical results show that the proposed network architecture outperforms the existing integrated networks.