Thanks for suggestion, Gordon. Here is the report from this evening. I hope I won't be the only one who does not have this working. :)

Thunderbird version 0.9+ (20041126), Palm Sync (2004-11-10), Contacts Sidebar 0.4.

Removed previous installation of PalmSync with /u and manually deleted the files, following the instructions at http://forums.mozillazine.org/viewtopic.php?t=128538&highlight=palm, posted by cd_racer. Removed Contacts Sidebar. Restarted Thunderbird a couple of times during these uninstallations.

Installed Palm Sync through extensions window. Set all the conduits except Mozilla Address Book to do nothing. Set MAB conduit to Handheld overwrites Desktop. Started Thunderbird. Synced the Palm (Visor Neo). The synchronization hanged, displaying the message with Switch to and Retry buttons. After  a couple of Retries and Switch to-s, the sync finished. No traces in log file. MAB conduit was reset to Synchronize the files. Edited one record in Contacts.

Disabled other conduits again. Resynchronized. MAB sync window was displayed for ten seconds or so. Couldn't find any traces of synchronization in the address book, though.

Added a new contact in the Palm address book. Set Handheld to overwrite Desktop. Disabled other conduits and resynchronized. It took a bit longer this time. Still no traces of sync in the address book(s).

Installed Contacts Sidebar 0.4, restarted Thunderbird. Disabled other conduits, set Handheld overwrites Desktop for MAB. Resynced.

Wohoo! The contacts were there in the Thunderbird Address Book.

Then I removed the Contacts AB that was imported from Outlook. Deleted the test contact in Thunderbird. Resynchronized all the conduits this time. The test contact was removed from Palm. Title I entered in one of the Thunderbird contacts was synchronized to the appropriate Palm contact. Collected Addresses AB was copied to Palm. Personal (address book) was copied to Palm.

Edited a record on Palm. The text was synced to Thunderbird AB record.

It works! :)

Alen