I went to the Oracle website for Java, now that they are the new owners of Java, and did some reading. I chose Java 6, and downloaded the JRE. They provide both RPM and self-extracting binary versions. I got the non-RPM version. I read the license agreement, yes really, before downloading it.
They have good installation instructions, and I installed it under /usr/local and set up symbolic links in /usr/local/bin from the jre1.6.0_33/bin executables. Then I tried aTunes, and yes, it now works.
There are specific instructions for manually installing the Java plugins for Firefox, but they don't have much to do with reality. As root, I went to /usr/local/mozilla/plugins and then created a symbolic link to /usr/local/jre1.6.0_33/plugin/i386/ns7/libjavaplugin_oji.so. Once that was in place, I restarted Firefox but still did not have Java support. I tried the other plugin directory /usr/local/jre1.6.0_33/plugin/i386/ns7-gcc29 but that still did not work.
I abandoned Java 6 and downloaded and installed Java 7. aTunes still works. The plugin instructions again do not match reality and I still have no Java plugin support in Firefox. I don't think I need that for anything, so I am going to leave it for the time being.
Our big Java application at work used to run on Linux and Mac, but when the sole developer did his last big upgrade a few years ago, he abandoned any pretense of platform interoperability and made the application Windows only. I think that was the only thing I needed the Java plugin for, and it's not available now so I don't care about the plugin.