Thanks, guys. After posting my message, I re-registered the software, got a new registration key, and the updates are now working fine.
It appears that updates are supposed to expire after about 1 year (see
http://www.avast.com/i_kat_207.php?lang=ENG ). I'm not sure why the "About" box lists different expirations for the software ("never") and the updates (a specific date), since updates are critical for proper operation of the program.
The new email from Avast clarifies that the registration is good for 12 months and will have to be renewed. (The earlier email from Avast, in late 2004, made no mention of renewal.) After entering the registration key, the "About" box shows an update expiration date 14 months from now.
It might be helpful for Avast to send out email messages to registered users shortly before their scheduled expiration dates, reminding them that they need to renew. Also, I'd suggest rewording the registration information in the "About" box, to eliminate the "Expiration: Never" line -- which would serve to emphasize that update eligibility will expire and registration needs to be renewed.
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I should note that the registration comment is a very minor point. I think Avast! is an excellent program, and I've been quite happy with it. Also, this support forum is very helpful.