Are you copying the places.sqlite file into your active profile folder? You can open your active profile folder this way:

Help > Troubleshooting Information > "Open Containing Folder" button

Close Firefox before copying the file in so it isn't overwritten when Firefox closes.

Your old profile folder also should contain a folder named bookmarkbackups. You can restore a JSON file from that folder. See this article for more information: Backing up and restoring bookmarks | How to | Firefox Help.

Thanks for responding, jscher. When I try to past it with firefox closed, I'm told it can't be done because the file is still in use somewhere else.

Firefox should release the file when it is shut down completely. If you are familiar with the Windows Task Manager (Ctrl+Shift+Esc), you could check the Applications and Processes tabs for Firefox and shut it down if you find it malingering there. Alternately, restarting Windows usually ensure that Firefox is not running.