Fluid-fluid phase separation in a simple model for a colloid dispersed in a solution of stiff rodlike polymers is studied by means of Gibbs ensemble Monte Carlo simulations. The fluid-solid transition for this model is computed using first-order perturbation theory. The phase diagrams clearly show that a stable critical point is not possible for polymer lengths smaller than 30% of the colloidal particle diameter. At very large rod length, the fluid coexistence curve can be derived directly from the hard-sphere equation of state.