Problems Running A Sketch on UNO with a 12 VDC external power supply

I have a simple sketch that runs fine on the board while it is plugged into the computer via USB, but when I connect a 12 VDC power supply and disconnect the board from the computer, the board doesn't even turn on (ie the 'ON' light remains off).

The power supply itself is supplying power to the board (the terminals read 12 VDC).

Confounded.

Does smoke come out?

Where do you plug the power supply exactly? Which pin are you using as the positive one?

No smoke, the power supply is plugged into the regular external supply port (the supply is a 12VDC, 1 A AC converter bought from RadioShack).

It has worked with other boards, but the two I have now are behaving the same.

Is the polarity correct? With your multimeter's positive lead on Vin and your negative lead on GND, you are seeing positive 12V?

I think I've resolved the problem (with a little help from http://www.reddit.com/r/arduino/) - the voltage regulator on the board is faulty. When tested, there is no voltage registered at the input and maybe 0.5 v registered at the output.

Also, there is 0 v registered between GND and Vin.

It seems it's time for another set of new boards o_0

GraemeOh: Also, there is 0 v registered between GND and Vin.

If that the case, then why would you expect the regulator to work? There's no voltage going into the regulator.