After much research and wringing of hands and many wrong turns in adapters, I finally have a DVI to ADC adapter coupled with a VGA to DVI connector to give the Cinema Display its own power and connect it to the VGA port of the G4. This did not work. Not a flicker, just a black screen. When I touch the display's power button, it does light up for a moment but doesn't stay on. From this, may I conclude that it is definitely not the power supply, or could it be that it just doesn't work with the VGA port?
So, I am likely left with the video card as the problem. I looked on the Apple website for the display model (M8537ZM/A) and found that 7 video cards supposedly work with the G4 and Cinema Display: NVIDIA GeForce2 MX, GeForce 3, GeForce 4 MX, GeForce 4 Titanium graphics card, and ATI Radeon 7500, 9000 Pro, or 9700 Pro graphics card. So my next question is which one should would be best for me to search for? I have no idea what is in there now. Before I start throwing even more money at this problem, I want to be sure I'm buying the right thing.