The temperature dependencies of the gamma(f.c.c.)-Ni/gamma'-Ni3Al(L1(2)) interfacial free energy for the {100}, {110}, and {111} interfaces are calculated using first-principles calculations, including both coherency strain energy and phonon vibrational entropy. Calculations performed including ferromagnetic effects predict that the {100}-type interface has the smallest free energy at different elevated temperatures, while alternatively the {111}-type interface has the smallest free energy when ferromagnetism is absent; the latter result is inconsistent with experimental observations of gamma'-Ni3Al-precipitates in Ni-Al alloys faceted strongly on {100}-type planes. The gamma(f.c.c.)-Ni/gamma'-Ni3Al interfacial free energies for the {100}, {110}, and {111} interfaces decrease with increasing temperature due to vibrational entropy. The predicted morphology of gamma'-Ni3Al(L1(2)) precipitates, based on a Wulff construction, is a Great Rhombicuboctahedron (or Truncated Cuboctahedron), which is one of the 13 Archimedean solids, with 6-{100}, 12-{110}, and 8-{111} facets. The first-principles calculated morphology of a gamma'-Ni3Al(L1(2)) precipitate is in agreement with experimental three-dimensional atom-probe tomographic observations of cuboidal L1(2) precipitates with large {100}-type facets in a Ni-13.0 at.% Al alloy aged at 823 K for 4096 h. At 823 K this alloy has a lattice parameter mismatch of 0.004 +/- A 0.001 between the gamma(f.c.c.)-Ni-matrix and the gamma'-Ni3Al-precipitates.