The development of Internet Hospital attracts growing attention worldwide to improve medical service quality and efficiency. However, the existing Internet Hospital failed to fully allocate the patients' consultation demands and improve the level of satisfaction, which is mainly caused by an overlong online waiting time during the diagnostic process. The emergence of Large-Language-Model (LLM) technology provides an opportunity to improve the existing synchronous and sequential online diagnostic process towards an asynchronous and concurrent process. With the consideration of applying LLM technology in developing the Internet Hospital, a conversational-AI-enhanced intelligent diagnostic process framework is proposed in this paper. By hierarchically decomposing the online diagnostic service into three layers, namely the AI doctor, the rotating doctor, and the expert doctor, the diagnostic process is capable of providing instant treatments with a lower misdiagnosis rate, meanwhile relieving the workload of human doctors. In addition, a case study of the Internet Hospital operated by Jiangsu Provincial Hospital is conducted, which reveals the importance of boosting the diagnostic progress by the proposed framework. Further, the proposed framework is examined by a numerical experiment based on statistical data. The experiment results indicate that both the patient waiting time and the misdiagnosis rate can be significantly reduced, which suggests a great potential for applying the proposed framework in practice.