The H atoms inside minihalos (i.e., halos with virial temperatures T-vir <= 10(4) K, in the mass range roughly from 10(4) to 10(8) M-circle dot) during the cosmic dark ages in a Lambda CDM universe produce a redshifted background of collisionally pumped 21 cm line radiation that can be seen in emission relative to the cosmic microwave background (CMB). Previously, we used semianalytical calculations of the 21 cm signal from individual halos of different mass and redshift and the evolving mass function of minihalos to predict the mean brightness temperature of this 21 cm background and its angular fluctuations. Here we use high-resolution cosmological N-body and hydrodynamic simulations of structure formation at high redshift (z greater than or similar to 8) to compute the mean brightness temperature of this background from both minihalos and the intergalactic medium (IGM) prior to the onset of Ly alpha radiative pumping. We find that the 21 cm signal from gas in collapsed, virialized minihalos dominates over that from the diffuse shocked gas in the IGM