The built in USB on the Arduino boards, are really a serial port in disguise.
So seen from Flash on your PC it will look like a serial port (COM port).
When you download and install the Arduino software on your PC, there's also a driver that makes it possible to use the "simulated serial port" .
So yes it is possible, and for the comunication alone you need nothing besides the USB cable.
This is also true if you want to communicate with other environments than Flash. For instance MAX/MSP.