A new fit to solar neutrino data in models with large extra dimensions

String inspired models with millimeter scale extra dimensions provide a natural way to understand an ultralight sterile neutrino needed for a simultaneous explanation of the solar, atmospheric and LSND neutrino oscillation results. The sterile neutrino is the bulk neutrino ($\nu_B$) postulated to exist in these models, and it becomes ultralight in theories that prevent the appearance of its direct mass terms. Its Kaluza-Klein (KK) states then add new oscillation channels for the electron neutrino emitted from the solar core. We show that successive MSW transitions of solar $\nu_e$ to the lower lying KK modes of $\nu_B$ in conjunction with vacuum oscillations between the $\nu_e$ and the zero mode of $\nu_B$ provide a new way to fit the solar neutrino data. Using just the average rates from the three types of solar experiments, we predict the Super-Kamiokande spectrum with 73\% probability, but dips characteristic of the 0.06 mm extra dimension should be seen in the SNO spectrum. We discuss both intermediate and low string scale models where the desired phenomenology can emerge naturally.