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The mailbox on your desktop is the application, put it back in the Applications folder.
 
I tried that already, but it creates a whole new one when I drag it in. Am I doing something wrong?

No. It is just making a copy. You can hold Command when dragging it to move it, or you can drag it there then delete the one from your desktop.

Note: If the icon in your dock doesn't work after doing this - drag it off, then open the app from the Applications folder. When the new icon appears in the dock, right-click and choose Keep in Dock ...
 
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