How can I measure the conductivity of a copper rod?
Very low resistance resistors need to be measured differently from how most ohmmeters work.  This is because the resistance of the meter leads would be significant compared to the measured resistance.

The solution is "4 wire" measurement.  Two wires are connected to the ends of the rod and are used to run a known current thru it.  This may be a few Amps.  Then make two more connections near the ends of the rod and measure the voltage accross them.  The resistance of these wires won't matter since they have very little current thru them.  The first set of wires may have more resistance than the copper rod.  This means they drop more voltage than the rod, but that doesn't matter since you are measuring the voltage only accross the rod.  Any extra voltage the power supply sees due to the connection wires is irrelevant as long as the total is low enough for the power supply to reliably supply the known current.

Given the current thru the copper rod and the voltage accross it, you can compute its resistance from Ohm's law.