You can archive your Facebook conversation on the Facebook web version (PC) and Messenger app (Android) and here's how:
Facebook Web Version (PC)
That's all. The conversation is successfully archived.
To view archived conversations: Click this URL, and you can see archived conversations.
Sending a message from or receiving a message to the archived conversation will automatically unarchive that conversation.
Facebook Messenger app (Android)
That's it. All done.
To view the archived conversation: Enter the person's name whose conversation you've archived in the search box, and you can see the archived conversation.
Send a message from or receive a message to the archived conversation will automatically unarchive conversation.
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How to Archive Facebook Conversation