One electrode? I don't know how that works, normally you would use two electrodes and measure the voltage between them. The voltage drop across the known resistor tells the current, you need the voltage between the two electrodes to determine the conductivity. Also be aware of any chemical reaction between the electrodes and the liquid that is being measured.
If you measure the voltage drop across the known resistor you can calculate the current that is flowing across the solution. The voltage drop across the solution is the applied voltage minus the voltage drop across the known resistor. Knowing the current and voltage you can calculate the resistance imposed by the solution. There is a small (but maybe significant) current flowing in the measuring instrument - you may need to account for that.
Realize that the voltage drop across the known resistor must be measured. It's not clear from the arrangement shown that ground or earth is the same for both the test apparatus and the measuring instrument.