No, all water has a small spontaneous rate of generating ion pairs. There
is a fresh supply of ions generated as long as there is water, so it is
only when the water all leaves the system that conduction stops.
What an electric current does, is shift the equilibrium concentration
due to that spontaneous generation, so that there is excess (OH-) ions
around the positive terminal, and excess (H3O+) ions around the negative
terminal. There may (or may not) be a reaction AT those terminals,
creating some chemical substances other than water and the electrode material.
Some generation of peroxide or hydrogen gas is certainly possible.
The concentration of those ions is represented by the pH of the water: near
the positive terminal (anode) the water will be basic, and near the negative
terminal it will be acidic. When current is stopped, equilibrium quickly
will be reestablished, as the charged ions repel their similarly-charged
In the case of AC current, there may be no non-reversible chemical changes
at the electrodes, just a mess of oddly-charged ions which revert to
neutral water, or there might be evolution of hydrogen and oxygen (and loss
Salt raises some of the rates, and adds to the possible reactions, but
is not essential: the ionizing solvent, water, has its own conductivity.