Bulk icosahedral-quasicrystalline aluminum-palladium-rhenium alloys of high structural quality and thermal stability are found to exhibit low-temperature electrical resistivities that are four orders of magnitude larger than those found in disordered metals and metallic glasses. Experiments suggest that these quasiperiodic alloys, which have a semimetallic electron density, are insulators at low temperature. The findings are discussed in light of theories on electron localization and band-gap formation in ordered metallic systems.