Nucleation and growth of thin films of Ag on the fivefold surface of an Al₇₂Pd_19.5Mn_8.5 icosahedral quasicrystal is studied with scanning-tunneling microscopy. For low coverages, flux-independent island nucleation is observed involving adatom capture at “traps.” With increasing coverage, islands start growing vertically, but then spread, and ultimately form hexagonal nanocrystals. These have fcc symmetry and pyramidlike multilayer stacking along the 〈111〉 direction. The constituent hexagonal islands have five different orientations, rotated by 2π/5, thus reflecting the symmetry of the substrate.