Pin 10, called "COM," of the ULN2803 is still connected to the Arduino's VCC
. That means that the cathodes of the ULN2803's internal freewheel diodes are all connected to +5V. But, the collector of your active Darlington driver is connected through the LED and resistor to +12V. So, when the input is low, the internal diode connected to pin 11 is forward-biased. Current flows from +12V, through the LED and the resistor to pin 11, and then through the internal diode to pin 10 at 5V. The LED stays on all the time. I'd expect it to dim and brighten a little bit as the input to the ULN2803 goes high and low.
For driving LEDs, the COM pin can be left unconnected, and that might be the best solution if you want to use different voltage sources to power the LEDs; otherwise, you'll always need to verify that COM is connected to the right voltage source. The purpose of the COM pin, and of the internal diodes that connect to it, is to provide a safe path for the energy stored in inductive loads, like relay coils or motor coils. Since you have none of those in the circuit, it's not necessary to connect COM to anything.
As for why the LED fades when you unplug the 12V power supply, it's likely that it's just energy stored in the capacitors of the supply. They discharge through the circuit, their voltage falls over time, and you see the LED fade to darkness. If you want to see the LED go dark quickly, try removing the +12V from the breadboard.
Edit: Here's a datasheet for the ULN2803: http://www.arduino-direct.com/Photos/uln2803a.pdf