A model is formulated for studying the shape of fast-rotating heated nuclei. Systematic calculations of the Gibbs-Routhian function FR, the Gibbs function F, and the energy E have been performed for rare-earth nuclei with 60 ⩽ Z ⩽ 76 and 90 ⩽ N ⩽ 108 in the interval of the frequency of rotation 0 ⩽ ω ⩽ 1 MeV for temperatures 0 ⩽ t ⩽ 2 MeV. It is found that at the temperature tcr ∼- 1-1.2 MeV corresponding to the excitation energy Ucr ∼- 14-17 MeV the shell effects become unimportant for the shape of rotating nuclei. The region of excitation energies and spins is found in which one expects large fluctuations of the shape. The density-of-states parameter for the rare-earth nuclei is found to be close to a = A/11.50 MeV - at t > tcr.