When you measure voltage, you have to measure the potential difference between two points, there is no "absolute" voltage.
You can run a second wire to ground and compare the voltages on the two wires. If the main wire is conducting any significant current, there will be a voltage drop across it proportional to the current times the resistance of the wire. Since the resistance of the wire should be very low, this voltage drop should be in the milli-volt range (it should really be zero as all the current should return through the neutral, not the ground).
This all assumes that you have standard multi-wire service (as opposed to single-wire, ground return which is a different beast).
If you are not particularly familiar with electronics, I would recommend using a cheap multi-meter with a data interface (preferably optical) to do the measurement. You can power the meter from a wall-wart (usually 9V, although some meters are 3 or 4.5V).
I suspect that all of this must be (by code) in an external box, I doubt you are allowed to add stuff to the meter box itself.
If you want to build something custom post another note and I'll post a schematic.
