MrGarak:
I think I remember reading something about people frying their uno r3 serial chips by using a separate power supply while having it plugged into USB.
I read them as well, but if your board works as expected when powered by USB only this shouldn't be the issue. Actually the reset may be caused by the failure on serial communication, i.e. whenever you open a serial communication (e.g., the serial monitor) the Uno resets (this is normal). So it may be an issue related to your PC - maybe you are using a baud rate that causes some instability, the PC resets the COM port and this in turn resets the Arduino. I don't know however how to investigate this. Maybe some message in the system logs?