Accurately projecting atmospheric levels and climate impacts of H2 requires knowledge of its biogeochemistry on time scales longer than the modern instrumental record. H2 trapped in polar ice is a potential archive for atmospheric H2 on millennial time scales, but ice core records of H2 do not exist because its high permeability in ice causes ice cores to rapidly exchange with the modern atmosphere. This work presents an analytical system capable of measuring H2 and neon in polar ice cores in the field immediately after drilling. This method has enabled the first ice core record of H2, and may help us better understand atmospheric H2 levels in the past.