Highly aligned nematic phases arise in systems of charged rodlike molecules or in highly concentrated solutions of uncharged rods. A multivariate Gaussian distribution is used to represent the orientation distribution of the rods and obtain predictions for the structure and rheology of the nematic solution. In weak flows, the director tumbles in one of a set of paths similar to the Jeffery orbits of a rigid rod with an effective aspect ratio equal to the reciprocal of the angular spread of the orientation distribution. The director migrates slowly across the orbits. An analytical expression for the rate of migration in the limit of small rotary Peclet number is obtained. The director may spiral toward the vorticity axis or the flow-gradient plane of the simple shear flow depending on its initial orientation. As the angular spread of the rods decreases, the domain of attraction of the vorticity axis becomes small.