One concern regarding the used nuclear fuel containers proposed for use in a Canadian deep geological repository (DGR) is the possibility that a small amount of hydrogen might be absorbed into their copper coating, potentially altering its mechanical properties. Reported herein is a study of hydrogen absorption into 50 nm of copper, coated on 4 nm of Ti using in situ neutron reflectometry (NR) and electrochemical impedance spectroscopy (EIS). NR results show that hydrogen is absorbed when the copper is cathodically polarized below the threshold for the hydrogen evolution reaction (HER), but that the hydrogen concentrates in the underlying titanium layer rather than concentrating in the copper coating. The hydrogen concentration in titanium rapidly rose when the HER was initiated and was observed to reach a steady state at TiH_1.5. Over the course of 55h of cathodic polarization, the concentration of hydrogen in the copper remained below the NR detection limit (2 at %). The portion of hydrogen atoms produced that diffused through the copper layer was initially 3.2%, suggesting a possible upper limit for hydrogen uptake by the copper coating of the UFC, although definitive conclusions can only be drawn from studies on 3 mm copper-coated steel samples.