An ultrathin film with a periodic interlayer spacing was grown by the deposition of Cu atoms on the fivefold surface of the icosahedral Al70Pd21Mn9 quasicrystal. For coverages from 5 to 25 monolayers, a distinctive quasiperiodic low-energy electron diffraction pattern is observed. Scanning tunneling microscopy images show that the in-plane structure comprises rows having separations of S=4.5±0.2  Å and L=7.3±0.3  Å, whose ratio equals τ=1.618… within experimental error. The sequences of such row separations form segments of terms of the Fibonacci sequence, indicative of the formation of a pseudomorphic Cu film.