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The structure of a capacitively coupled high-pressure glow (HPG) discharge in high-purity helium is investigated using a detailed one-dimensional modeling approach. Impurity effects are modeled using trace amounts of nitrogen gas in helium. Average electron temperatures and densities for the HPG discharge are similar to their low-pressure counterpart. Helium-dimer ions dominate the discharge structure for sufficiently high-current densities, but model impurity nitrogen ions are found to be dominant for low-discharge currents. Helium dimer metastable atoms are found to be the dominant metastable species in the discharge. The high collisionality of the HPG plasma results in significant discharge potential drop across the bulk plasma region, electron Joule heating in the bulk plasma, and electron elastic collisional losses. High collisionality also results in very low ion-impact energies of order 1 eV at the electrode surfaces.