The temporal evolution of ordered γ'(L12 structure)-precipitates precipitating in a disordered γ(f.c.c. structure)-matrix is studied for Ni-12.5 Al and Ni-13.4 Al at.% alloys aged at 773, 798, 823, or 873 K for times ranging from 0.08 to 10,287 h in a detailed and comprehensive study utilizing three-dimensional atom-probe tomography (3-D APT), transmission electron microscopy (TEM) and monovacancy-mediated lattice-kinetic Monte Carlo (LKMC1) simulations.  The 3-D APT results are compared in detail to LKMC1 simulations, which include monovacancy-solute binding energies through 4th nearest-neighbor distances, for the same mean compositions and aging temperatures. The temporal evolution of the measured values of the mean radius, number density, aluminum supersaturation, and volume fraction of the γ'(L12)-precipitates are compared to the predictions of a modified version of the Lifshitz-Slyozov  and Wagner  coarsening model due to Calderon, Voorhees, Murray and Kostorz , which takes into account the thermodynamics of the γ(fc.c.)-matrix. The resulting experimental rate constants are used to calculate the Gibbs interfacial free-energies between the γ(f.c.c)- and γ'(L12)-phases using data from two thermodynamic databases, and its value is compared to archival values dating from 1966. The diffusion coefficient for coarsening is then calculated from the same rate constants and is compared to all extant archival diffusivities, not determined from coarsening experiments, and is demonstrated to be the interdiffusivity, D∼, of Ni and Al at the four aging temperatures. The monovacancy-mediated LKMC1 simulation results are found to be in good agreement with 3-D APT data for all the quantities determined by 3-D APT. Furthermore, it is demonstrated that the compositional interfacial width, for the (100) interface, between the γ(f.c.c)- and γ'(L12)-phases, decreases continuously with increasing aging time and mean radius for the 3-D APT results and the monovacancy-mediated LKMC1 simulations , in disagreement with the ansatz that is intrinsic to the trans-interface diffusioncontrolled coarsening model, which predicts the opposite trend for binary Ni-Al alloys .