Using electron diffraction, we show that free-electron metals, such as sodium and potassium, form a highly regular quasiperiodic monolayer on the fivefold surface of icosahedral Al-Pd-Mn and that the quasiperiodicity propagates up to the second layer in sodium. Our photoelectron spectroscopy results show that the quasicrystalline alkali-metal adlayer does not exhibit a pseudogap near the Fermi level thought to be characteristic for the electronic structure of quasicrystalline materials. Calculations based on density functional theory provide a model structure for the quasicrystalline alkali-metal monolayer and confirm the absence of a pseudogap.