We report results from a program of near-infrared spectroscopic observations of the H2 emission from planetary nebulae, being carried out at McDonald Observatory using an InSb array-detector spectrometer. Our observations employ both high spatial resolution (3″ diameter aperture) and high spectral resolution (λ/Δλ = 200–600), thus avoiding potential problems with line blending and spatial registration. These observations provide simultaneous measurements of H I recombination lines and H2 emission lines, thus accurately defining the relative extent and distribution of the ionized vs. molecular material. One-dimensional cuts through the compact planetary nebulae BD+30 3639 and Hubble 12, taken along east-west and north-south axes through the nebular centers, show that the H2 emission is concentrated in a ring or shell outside the ionized nebular core. The angular extent of the H2 emission in Hb 12, with a characteristic diameter of about 8–10″ arc seconds, is strikingly larger than the dimensions of the ionized core, which is less than 2″ in diameter.