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The primary goal of the Genesis Mission is to collect solar wind ions and, from their analysis, establish key isotopic ratios that will help constrain models of solar nebula formation and evolution. The ratios of primary interest include 17O/16O and 18O/16O to ±0.1%, 15N/14N to ±1%, and the Li, Be, and B elemental and isotopic abundances. The required accuracies in N and O ratios cannot be achieved without concentrating the solar wind and implanting it into low-background target materials that are returned to Earth for analysis. The Genesis Concentrator is designed to concentrate the heavy ion flux from the solar wind by an average factor of at least 20 and implant it into a target of ultra-pure, well-characterized materials. High-transparency grids held at high voltages are used near the aperture to reject >90% of the protons, avoiding damage to the target. Another set of grids and applied voltages are used to accelerate and focus the remaining ions to implant into the target. The design uses an energy-independent parabolic ion mirror to focus ions onto a 6.2 cm diameter target of materials selected to contain levels of O and other elements of interest established and documented to be below 10% of the levels expected from the concentrated solar wind. To optimize the concentration of the ions, voltages are constantly adjusted based on real-time solar wind speed and temperature measurements from the Genesis ion monitor. Construction of the Concentrator required new developments in ion optics; materials; and instrument testing and handling.