A new method of solution of the Einstein field equations is used to investigate the gravitational collapse of a rotating massive body. The equations are studied on a timelike hyperplane of reflection symmetry within an axially symmetric distribution of perfect fluid. The assumptions of negligible pressure and uniform angular velocity in the hyperplane, as well as certain restrictions on the development of the metric off the hyperplane, allow the metric to be reduced to a simple form. An explicit family of solutions is obtained for the evolution in time of the metric at a fixed radius in a comoving coordinate system. These solutions contain as limiting cases, valid throughout the hyperplane, the nonrotating collapse of a uniform-density sphere and the Newtonian equation of motion for a class of finite rotating bodies. The general solution reveals that rotation is unable to halt gravitational collapse at a given comoving radius if the ratio of rest mass "interior" to that radius to angular momentum per unit mass at that radius is sufficiently large. In that case the contribution of the rotational kinetic energy to the effective gravitational mass becomes more important than the opposing effect of centrifugal force. This leads to a dynamical singularity, in which the proper-time derivatives of both radial and circumferential distances become negatively infinite, while the distances themselves remain finite.