We have studied the effect of irradiation with 900 MeV Pb and 780 MeV Xe ions on quasicrystals Al62Cu25.5Fe12.5 and the related cubic α-phase Al55Cu27Fe11Si7. The fluences ranged from 1010 to 5.8 × 1013 ions/cm2. Irradiations were performed at 80 K and at room temperature. The structural changes induced by the electronic excitations were studied by high-resolution X-ray diffraction. Whereas one might have expected the irradiations to induce important structural transformations, e.g. a phase transition from an icosahedral to a rhombohedral phase, only minor structural modifications are observed in the icosahedral phase and none in the α-phase. The defects created are not phasonic. Our results suggest a remarkable structural stability of these phases with respect to heavy-ion irradiation.