The thread on the Java Plug-In forum on java.sun.com describing the problem, in particular a link to a test case, is http://forum.java.sun.com/thread.jspa?threadID=5230759&start=30 -- look for the posting by Cyrille linking to http://oueb.org/AppletFireFoxLiveconnectBug.html . A copy of this test case will be made internally and described in the Comments section.
The problem appears to be that some code in the Java Plug-In resolves hostnames to IP addresses where it used to be naming the host symbolically. Access control checks are failing because on one side we have the symbolic host name and on the other side its IP address, and the security checks do not understand that these are equivalent.
Note that we copied this test case in house and it does not fail when hosted on our internal servers. The issue appears to be related to how name service lookups work for the particular host.