The first stars in the universe, forming at redshifts z >= 15 in minihalos with masses of order 10(5-6) M-circle dot, may leave behind black holes as their remnants. We discuss how these black holes can grow through accretion and what their radiative feedback effects are on the surrounding medium. They may also serve as "seeds" for the much larger black holes observed at redshifts z <= 7. To investigate their fate, we have used the adaptive mesh refinement code Enzo to carry out simulations of the formation, radiative feedback, and evolution of the relic HII region of a 100 M-circle dot population III star. We explicitly track the position of the black hole it leaves behind, which allows us to estimate the accretion rate and associated radiative feedback effects. We find that although the black hole's mass does not change substantially for the first few hundred million years due to radiative feedback from its progenitor, photoionization heating from accretion is likely to have a strongly self-regulating effect. This will likely limit accretion onto these black holes in minihalos, even when they encounter cold, dense gas and fall into larger dark matter halos.