Well, it's a sad story.
Mozilla/Firefox completely removed support for java applets in web browser. Some for Chrome. Instead they suggest to switch to "java webstart technology". I don't know yet how difficult it is to switch, I will have to look into it.
With Internet Explorer I managed to run the game, but it requires some tinkering.
1. In Internet Explorer, go to Internet Options/Security and use the slider to change security level to "Medium" (medium-high is not enough)
2. In windows Control Panel, find java control. E.g. in Windows 7 click Start button / Control Panel and type java in search box. OR in Control Panel, click "Programs" and the java control should be there.
Once in java control, go to "security" tab, and add http://duelboard.com
to exception site list. You need to add whole path, including /old
Then it works.
BTW I am impressed somebody still cares about CB at least little bit.
Now I see, while the game loads, it still won't work. It cannot connect to the server. Due to some increased security restrictions for java. I'll try to investigate what can be done about it.