Except I don't believe any of the ADC to DVI adapters pass the analog signal through. It may be possible to hack your own, though.
So the problem lies in the fact that ADC was Apple's custom adapter that was really a DVI-I plug plus USB and power. DVI-I, as opposed to DVI-D, carries both the digital signal that is usually associated with DVI, as well as the stock VGA analog signal.
Every single ADC to DVI adapter I've ever seen specifies that it only works with the LCD displays, which would mean that it's only passing through the digital signal, not the analog signal. (The CRT ADC display is the only one that used an analog signal.)
Again, if you're adventurous, you may be able to hack an ADC-DVI adapter so that you're only using it for power, and manually re-wire it so that you get the analog video signal out of it, though.
But I don't know if I'd be willing to try to do that...