As I sit here typing this, that's exactly what I have running in front of me....
A 9v nominal, 8.3V actual (how's that for coincidence?) PP3 plugged into the Uno barrel plug. The servo gets its own power from 4x AAs and the grounds are coupled.
All working as expected.
You sure you have the battery polarity the right way round- centre positive.
So I've done my research and I'm ready to connect the Arduino to an external power source. The battery is giving off around 8.3V, I've read that the Arduino can take 7-12V, however, when I plug in the battery's power jack to start the board, nothing happens.
Yes, that is all I can think of, polarity of your 8.3vdc power plug is wrong for an arduino? Must be center pin positive and if reversed there is a polarity protection series diode to prevent damage to the arduino board, but would give you the symptom you see, nothing works and the board's power led would be off.
Don't thank me, thank the Arduino guys who put that anti-fry diode in the Uno 8)