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Strong field-confinement, long-lifetime resonances, and slow-light effects suggest that meta surfaces are a promising tool for nonlinear optical applications. These nanostructured devices have been utilized for relatively high efficiency solid-state high-harmonic generation platforms, four-wave mixing, and Raman scattering experiments, among others. Here we report the first all-dielectric metasurface to enhance harmonic generation from a surrounding gas, achieving as much as a factor of 45 increase in the overall yield for Argon atoms. When compared to metal nanostructures, dielectrics are more robust against damage for high power applications such as those using atomic gases. We employ dimerized high-contrast gratings fabricated in silicon-on-insulator that support bound states in the continuum, a resonance feature accessible in broken-symmetry planar devices. Our 1D gratings maintain large mode volumes, overcoming one of the more severe limitations of earlier device designs and greatly contributing to enhanced third- and fifth- harmonic generation. The interaction lengths that can be achieved are also significantly greater than the 10's of nm to which earlier solid-state designs were restricted. We perform finite-difference time-domain simulations to fully characterize the wavelength, linewidth, mode profile, and polarization dependence of the resonances. Our experiments confirm these predictions and are consistent with other nonlinear optical properties. The tunable wavelength dependence and quality-factor control we demonstrate in these devices make them an attractive tool for the next generation of high-harmonic sources, which are anticipated to be pumped at longer wavelengths and with lower peak power, higher repetition rate lasers.