Abstract.The equation of the Bohr-Mottelson Hamiltonian with a sextic oscillator potential is solved for γ-rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the β equation is brought to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The 0 + and 2 + states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental β bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate β-soft rotor crossing through a critical point. Numerical applications are performed for