We report on studies of accretion of material of solar composition at a variety of rates onto hot, massive white dwarfs (WDs). Evolutionary sequences for 1.25 M⊙ and 1.35 M⊙ WDs with mass accretion rates varying from 1.6 × 10-10M⊙yr-1 to 3.5 × 10-7M⊙ yr-1 have now been studied and, in all these cases, tlie WDs grow in mass to near the Chandrasekhar Limit. The surface conditions of our simulations are similar to observations of Super Soft X-ray binary sources. Our results provide the first confirmation of previous suggestions that some of the Super Soft X-ray binaries may be the progenitors of thermonuclear supernovae. We observe that the abundances in the accreted layers (including carbon and oxygen) are sensitive to the accretion rate, potentially affecting the dynamics of the supernova explosion and observations of elemental distributions.