I searched high and low for evidence that someone else out there had run into this particular problem, but I was unable to find anything. So I am writing this in the hopes that the next person who faces this particular problem will have an easier time.
I had just used Picasa to import about two thousand photos from memory cards onto my new Windows 7 computer. I then noticed that these files were stored in C:\Users\Jake\Pictures, not the "My Pictures" folder like I expected (On my system I set up "My Pictures" to be on a larger drive since my boot drive is a relatively small SSD). I changed the location in Picasa's settings, but this didn't move any of the files it had already imported.
I closed Picasa and copied the files to the My Pictures folder, then re-opened Picasa. Then I started to notice that every file now had a duplicate (both in Picasa and on disk). For whatever reason, Picasa created an extra copy of each file. For example, IMG_5435.JPG now had a twin, IMG_5435-1.JPG.
I searched high and low for an easy way to use Picasa to remove these duplicates. However, it does not appear that Picasa's search feature offers wildcard search. The way it handles characters like - and . and ? is a mystery to me. Searching for "1.jpg" will return results such as "IMG_3145.JPG".
So in the end I used Windows Explorer to find the duplicates, as it is happy to search for wildcards. Since all the duplicates ended in -1, I just searched for *-1.JPG and it showed the 1,982 duplicates. I then hit Ctrl-A to select all, and Del to delete them. Close and reopen Picasa, and boom. No more duplicates.