Here's some feedback on how it worked out using CB Juice (latest version, beta 10):
First, I had to make sure both installations included the exact same fields. I had deleted a field on the new installation which was still present on the old one, and I had to recreate it (with the exact same name) to import the user list from the old site.
CB Juice allows me to either add missing users, or update all users. I started by adding the missing users. That mostly worked, but the passwords were not carried over properly. I needed to reset those manually and have the members set them again.
Then I updated all users so that any profile changes they made while I was developing the new site would be carried over. That turned out not to be such a good idea. All changes got carried over fine, but each and every user's name (their real name, not their username) got wiped out. They all became blank. I was able to restore those from the database, but it took a long time.
All the users are carried over now, and we can launch the new site.