We describe a simulation method for the accurate study of the equilibrium freezing properties of polydisperse fluids under the experimentally relevant condition of fixed polydispersity. The approach is based on the phase switch Monte Carlo method of Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)]. This we have generalized to deal with particle size polydispersity by incorporating updates which alter the diameter σ of a particle, under the control of a distribution of chemical potential differences μ̃ (σ). Within the resulting isobaric semi-grand-canonical ensemble, we detail how to adapt μ̃ (σ) and the applied pressure such as to study coexistence, while ensuring that the ensemble averaged density distribution ρ (σ) matches a fixed functional form. Results are presented for the effects of small degrees of polydispersity on the solid-liquid transition of soft spheres.