The nuclear hyperfine tensor (A) components of the 2,2'-diphenyl-1-picrylhydrazyl neutral radical are computed using the UB1LYP hybrid density functional method. Solvent interactions via hydrogen bonding are found to play a crucial role in the position of the two phenyl rings relative to the picryl moiety. Under these conditions, the calculated isotropic hyperfine tensor components of the N-1 and N-2 hydrazyl backbone are within similar to 1.3 Gauss (G) of the experimental values determined by EPR and ENDOR spectroscopy. Just as important are the effects of restricted rotations of the phenyl rings on these tensors. Rotational averaging using a Maxwell-Boltzmann type distribution improves the agreement between theory and experiment to less than 1.0 G. In addition, rotational averaging of the twelve isotropic proton coupling constants has also been performed. They come within 0.3 G of the experimental values. Thus, for the first time, all the nuclear hyperfine tensor components of this large class of molecules are accurately calculated without resorting to post Hartree-Fock techniques.