<p>We'd like a trace in our application logs of these exceptions - by default Java just outputs them to the console.</p>

There are two ways:
/* Install a Thread.UncaughtExceptionHandler on the EDT */
Set a system property:
System.setProperty("sun.awt.exception.handler",MyExceptionHandler.class.getName());
I don't know if the latter works on non-SUN jvms.
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Indeed, the first is not correct, it's only a mechanism for detecting a crashed thread.