Growth conditions for the preparation of a single Al₆₄Cu₂₃Fe₁₃ icosahedral (I-) quasicrystal with excellent quasicrystallinity were examined using the Czochralski method. The appreciation of the quasicrystallinity of the grown single quasicrystal was performed by X-ray structural analysis. The full widths at half-maximum (FWHM) of the Bragg reflections along 2-, 3- and 5-fold symmetry directions have no Q_|| and Q_⊥ dependence. Where the Q_|| and Q_⊥ mean the phason momentum and real scattering vector. Furthermore, peak shifts from ideal Bragg positions were not observed. These means that the grown Al₆₄Cu₂₃Fe₁₃ quasicrystal by the Czochralski method has perfect I-phase structure.