Assuming you haven't made any changes to the library (or even the metadata within the library):
First, switch the library in Calibre from the original location to the new Dropbox location:
- Click on the Calibre Library toolbar button
- Choose "Switch/create library..."
- Navigate to the Dropbox location in the "New Location" box
- Leave the selection as "Use the previously existing library at the new location" button selected
- Click OK
That'll switch you to the new library in the Dropbox folder.
Then delete (or backup/move, just to be safe) the old library. Calibre will point to the new location and do any changes from there. You could also rename the old library copy (Calibre Library-old, for example) and leave it where it is and switch back and forth between them, if you're not sure the two libraries are exactly the same. Then you can delete the old one once you know you've made any necessary adjustments. (YOu can move and/or copy books from one library to the other).
I use Dropbox and have by Calibre library saved there. I would make the following suggestion as well: I turn off the Dropbox syncing most of the time and only run it once in awhile. However, if you want to leave it running most of the time, I'd suggest you turn it off before you open Calibre. If Dropbox tries to backup or copy your metadata.db file while you're working in it, it can cause problems.
Good luck!