The transition from a spherical-vibrator to a deformed-rotor phase is examined in this and succeeding papers in the interacting boson and Bohr–Mottelson collective models. This paper explores the transition from a U ( 5 ) to an O ( 6 ) phase in the interacting boson model with variation of a control parameter α. As expected from other studies, it is observed that a low-lying state of the model can be assigned to one of three phases: a U ( 5 ) phase, an O ( 6 ) phase, or a transition phase. It is also shown that the range of values of the control parameter, for which the low-lying states are in the transition phase, shrinks as the dimension of the IBM Hilbert space increases and that a sharp phase transition develops as the infinite-dimensional limit is approached. A detailed examination of the states of the two phases shows that the U ( 5 ) symmetry of the α = 0 limit is mixed throughout much of the U ( 5 ) phase and the O ( 6 ) symmetry of the α = 1 limit is even more strongly mixed throughout much of the O ( 6 ) phase. Nevertheless, it is found that quasidynamical U ( 5 ) and O ( 6 ) symmetries continue to characterize the phases and provide an explanation of the apparent persistence of these symmetries in the face of strong symmetry-breaking interactions. Thus, the analysis suggests a useful definition of a phase associated with a dynamical symmetry. It also adds a perspective to Landau's theory of second-order phase transitions and to Iachello's concept of a critical point symmetry with analytical solutions in the transition region.