David at Infinite Monkeys suggests the following to fix bloggered archives.

Go to settings and archiving, and turn off archiving. Blogger swears you won't lose anything, they are still archiving. Blogger also suggests that you now publish, then turn back on archiving. But don't do that. Just leave archiving off. Your permalinks on your main page will now work, and, strangely, your archives sit there untouched.

Then it gets weirder. Brad e-mailed me the next day and said "I see you fixed the archiving, because our August archive is working again." So, after a few hours, I decided to risk things and turn the archive back on. Everything worked fine.

And it worked for us, too. Thanks, David!