The ohmic resistance of a thin-film YSZ electrolyte was measured as a function of the electrolyte thickness using an Au paste electrode. A residual resistance was found when the resistance vs. thickness relation was extroplated to the zero thickness. It is independent of the oxygen pressure, and its activation energy is identical to that of YSZ resistivity. Assuming a square mesh contact, we calculated the effective resistance as a function of thickness. The calculated resistance agreed well with the experimental one, suggesting that the residual resistance is due to the electrode-electrolyte contact in discrete spots.