Most of the open-domain dialogue models tend to perform poorly in the setting of long-term human-bot conversations. The possible reason is that they lack the capability of understanding and memorizing long-term dialogue history information. To address this issue, we present a novel task of Long-term Memory Conversation (LeMon) and then build a new dialogue dataset DuLeMon and a dialogue generation framework with Long-Term Memory (LTM) mechanism (called PLATO-LTM). This LTM mechanism enables our system to accurately extract and continuously update long-term persona memory without requiring multiple-session dialogue datasets for model training. To our knowledge, this is the first attempt to conduct real-time dynamic management of persona information of both parties, including the user and the bot. Results on DuLeMon indicate that PLATO-LTM can significantly outperform baselines in terms of long-term dialogue consistency, leading to better dialogue engagingness.