Dilution of reacting species by build-up of helium ash and its effect on ignition in the ITER tokamak have been studied in a series of simulations with the one-dimensional BALDUR transport code. Thermal diffusivities, obtained from ITER scaling laws and with radial variations observed in JET, gave tau-E almost-equal-to 2-4 s. Refuelling of deuterium and tritium maintained constant electron density, while carbon recycling was 100% and the helium ash recycling was varied from 1.0 to 0.5. Including MHD effects, specifically sawteeth and beta limits, we find that ignition can be sustained for 200 s with R(helium) = 0.95. These simulations, the only non-zero-dimensional, time-dependent simulations thus far made for ITER plasmas, emphasize that edge plasma conditions, MHD behavior, and helium particle transport are critical synergistic issues for sustained ignition.