Tritium permeation through structural materials of fusion reactors is an important issue from both safety and tritium self-sufficiency points of view. Mutual influences among hydrogen isotopes are also important in assessments of tritium permeation fluxes, and although some theoretical and experimental works have been carried out so far—which have all focused on steady-state permeation—their conclusions are insufficiently clear about how one isotope affects the permeation of another. This was motivation to further investigate this problem, both theoretical and experimental in a non-steady-state approach and also in the surface-limited regime (SLR) of permeation. After initial theoretical work, a dedicated experimental installation for gas-driven permeation experiments was designed and assembled. Then, initial experimental work was completed for testing monoisotope permeation of deuterium through a very thin (0.075-mm) nickel membrane in the temperature range of 473 to 773 K and for a driving pressure of deuterium gas in the range of 10−2 to 1 Pa. This work was done in preparation for subsequent multi-isotope experiments, with the proposed goal to observe how hydrogen affects the permeation of deuterium. The main objectives of the work were to confirm experimentally the achievement of the SLR of permeation for the selected conditions of testing; to determine experimentally the kinetic coefficients of surface transport for deuterium in nickel; and to test and validate the experimental setup, procedures, and methods used and their reliability to more stringent requirements of the multi-isotope experiments. Within this paper, the experimental setup and all the operating procedures used both in calibration operations and in permeation experiments are presented in detail, as well as their results. The obtained results confirmed that permeation occurred in the SLR in the tested range of pressure for each testing temperature. The surface-rate coefficients of dissociation and recombination were both determined experimentally. The values obtained for the dissociation coefficient were in very good agreement with other similar experimental data available in the literature. For the recombination coefficient, agreement was not quite satisfactory, but comparison could be made with values calculated based only on the dissociation and solubility coefficients, as data for these coefficients for the nickel-hydrogen system (determined directly from gas-driven permeation experiments) were not found in the literature. However, these results indicated good reliability of the installation, its calibration, and operating procedures in order to proceed to experimental testing of multi-isotope permeation.