I ended up reinstalling OS X 10.8 and then installing the 10.8.5 Combo Update (I keep disc images of each system install and combo updates). That got me back to the proper (older) version of Java 6 that works with DD Poker 3:
java version "1.6.0_51"
Java(TM) SE Runtime Environment (build 1.6.0_51-b11-457-11M4609)
Java HotSpot(TM) 64-Bit Server VM (build 20.51-b01-457, mixed mode)
In Terminal (Applications>Utilities), type "java -version" and you'll see the version you're running.
If you're still using 10.6.x (Snow Leopard) or 10.7.x (Lion), I suspect that reinstalling back to the base versions of those systems and then installing the last Combo Update should do the trick, minding not to install any update to Java that the Apple Software Update wants you to do.