Apple have finally patched up some serious security holes in their Java distributions with Mac OS X. I had a problem when trying to install it, getting the simple message The update "Java for OS X 10.5 Update 4" can't be installed almost immediately after pressing the Install button (it didn't even start downloading). I'd just freshly started up my Mac and the browser wasn't running (it specifically instructs you to quit all browsers) so that wasn't the problem.
Repairing permissions on my startup disk with Disk Utility then restarting seemed to fix it and allow the installation to proceed. Now I have 1.6.0_13 successfully installed.
Update: Exactly the same experience on my PowerMac G5 (original attempt was on an Intel MacBook). Same problem, same solution. Could it be because I've previously used the Java Preferences app to switch between JVM versions (like you're supposed to)? It's just about conceivable that the way the prefs app performs the switching leaves permissions a bit messed up as far as the installer is concerned, but none of the permissions it claimed to have fixed looked to be related in any way. Maybe it was more the restart that helped. Certainly just fixing permissions alone was not enough.
Installed for me without a hassle.