We present the results of gas-phase NMR measurements designed to yield a new experimental value for the absolute H-1 magnetic shielding for an isolated hydrogen molecule and its deuterium isotopomers. The results are based on the original method of direct shielding measurements (Jackowski et al., 2010) and the density dependence of H-1, H-2, and He-3 NMR frequencies for molecular hydrogen and atomic helium-3. The absolute isotropic magnetic shielding measured for molecular hydrogen, sigma(0)(H-2), is 26.293(5) ppm at 300 K, within experimental error of previous measurements based on spin-rotation data and quantum chemistry computations, 26.289(2) ppm (Sundholm and Gauss, 1997), and recent ab initio calculations. We also report isotope effects in shielding for H-2, HD, and D-2 molecules that are consistent with theoretical predictions. In addition, gas-phase H-1 chemical shifts extrapolated to zero density have been measured for numerous small molecules. Our results yield precise absolute shielding data that will be useful in establishing benchmark computational chemistry methods for calculating rovibrational averaged magnetic shielding.