Just to add one more post with the same problem: Running v6.0.1289, on Windows XP Pro. My registration expires in ~25 days. I tried to re-register online, but the button underneath the "Base Protection" option is non-functional (same screen as what Rick F posted a few messages back). Just to check, I also tried to click on the "Full Protection" button, and that launched my browser, which I am guessing would eventually result in a successful paid registration. No browser was running when I first tried either button, later on I had Opera running, and the "Base Protection" button still didn't work (I didn't retry the "Full Protection" button again).
Anyway, I was able to successfully re-register using the offline/email method, but someone should look into fixing this online problem. Or if it's a server issue, then have some sort of message indicating as such and telling people to try later (even if it has to be 'pre-printed' on the screen). As it is now, it looks like an attempt to force people to upgrade to the paid option, and that doesn't reflect well on avast!