In order to ascertain the final masses of Population III stars, which are believed to be 30-300 M<SUB>solar</SUB>, we must follow the prolonged period of accretion during which these stars are believed to gain the majority of their mass. The key question is, essentially, what terminates accretion, and at what mass? In order to answer this question, we turn to 3D simulations with ionizing radiative feedback that are capable of following the flow for many accretion timescales. Since following ab-initio simulations including a fully hydrostatic core for such a long time will remain computationally infeasible for the foreseeable future, we use a sink particle approach to remove the highest density (n〉~10 cm) material. We describe our implementation of a ray tracing approach to ionizing radiation transport in our Orion AMR radiation hydrodynamics code.