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The highest priority of the Solar and Atmospheric Neutrino Experiment Working Group is the development of a real-time, precision experiment that measures the pp solar neutrino flux. A measurement of the pp solar neutrino flux, in comparison with the existing precision measurements of the high energy {sup 8}B neutrino flux, will demonstrate the transition between vacuum and matter-dominated oscillations, thereby quantitatively testing a fundamental prediction of the standard scenario of neutrino flavor transformation. The initial solar neutrino beam is pure {nu}{sub e}, which also permits sensitive tests for sterile neutrinos. The pp experiment will also permit a significantly improved determination of {theta}{sub 12} and, together with other solar neutrino measurements, either a measurement of {theta}{sub 13} or a constraint a factor of two lower than existing bounds. In combination with the essential pre-requisite experiments that will measure the {sup 7}Be solar neutrino flux with a precision of 5%, a measurement of the pp solar neutrino flux will constitute a sensitive test for non-standard energy generation mechanisms within the Sun. The Standard Solar Model predicts that the pp and {sup 7}Be neutrinos together constitute more than 98% of the solar neutrino flux. The comparison of the solar luminosity measured via neutrinos to that measured via photons will test for any unknown energy generation mechanisms within the nearest star. A precise measurement of the pp neutrino flux (predicted to be 92% of the total flux) will also test stringently the theory of stellar evolution since the Standard Solar Model predicts the pp flux with a theoretical uncertainty of 1%. We also find that an atmospheric neutrino experiment capable of resolving the mass hierarchy is a high priority. Atmospheric neutrino experiments may be the only alternative to very long baseline accelerator experiments as a way of resolving this fundamental question. Such an experiment could be a very large scale water Cerenkov detector, or a magnetized detector with flavor and antiflavor sensitivity. Additional priorities are nuclear physics measurements which will reduce the uncertainties in the predictions of the Standard Solar Model, and similar supporting measurements for atmospheric neutrinos (cosmic ray fluxes, magnetic fields, etc.). We note as well that the detectors for both solar and atmospheric neutrino measurements can serve as multipurpose detectors, with capabilities of discovering dark matter, relic supernova neutrinos, proton decay, or as targets for long baseline accelerator neutrino experiments.