Duemilanove consistently expires after around 60 seconds?

I'm holding one clip to the metal tab on the inside of the 9V barrel plug, and keeping the other attached to ground. Reading is constant around 6V

Either your battery is dead or you are drawing too much current from the battery. Given how little current a 9V can supply, this isn't a surprise.

I ditched the 9V battery plug and put 2 9V batteries in series, so now I've got 18V going into Vin.

You'd be better off putting them in parallel, doubling the current capacity of your supply. Doubling the voltage isn't going to fix the problem. You probably need to look at a different battery system. 9Vs are lousy. They are exceptionally lousy for motors.

Try calculating (and measuring) the amount of current your system is using.