Generally you only connect the Ground between two systems with different voltage requirements.
John, actually I'm ok using two power supplies. Are you saying that connecting the ground and the vin from the 6V device to the UNO while powering the UNO with an additional
9V somehow prevents an over-voltage flow to the device?
You should connect the Arduino Ground to the BPM Ground but don't connect the + side of the two power supplies together.
Try powering the BPM from Arduino Ground and Arduino +5V (NOT Vin). If the BPM will run on 5V that will make is safer for the Arduino. Six volts is the absolute maximum for an Arduino data pin.