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If you're logged into Facebook, doing so will open your News Feed. If you aren't logged into Facebook, enter your email address (or phone number) into the top-right corner of the screen and click Log In. It looks like a message bubble with a lightning bolt inside, in the row of options located in the top-right side of the Facebook page. This link is at the bottom of the Messenger drop-down menu. Clicking it will open the Messenger utility. Your conversations are stored on the left side of this page. You should see a small gear icon appear in the bottom-right corner of the selected message. Doing so will open a drop-down menu with several options. It's near the middle of the drop-down menu. You should see this option on the pop-up "Delete Conversation" window. Clicking it will permanently remove your selected conversation from your messages history.
Open the Facebook website. Click the Messenger icon. Click View All in Messenger. Scroll to the conversation you wish to delete. Hover your mouse cursor over the message. Click the ⚙️ icon. Click Delete. Click Delete when prompted.