  We find that at redshifts z > 10, HD line cooling allows strongly-shocked
primordial gas to cool to the temperature of the cosmic microwave background
(CMB). This temperature is the minimum value attainable via radiative cooling.
Provided that the abundance of HD, normalized to the total number density,
exceeds a critical level of ~ 10^{-8}, the CMB temperature floor is reached in
a time which is short compared to the Hubble time. We estimate the
characteristic masses of stars formed out of shocked primordial gas in the wake
of the first supernovae, and resulting from the mergers of dark matter haloes
during hierarchical structure formation to be ~ 10 M_{solar}. In addition, we
show that cooling by HD enables the primordial gas in relic H II regions to
cool to temperatures considerably lower than those reached via H_2 cooling
alone. We confirm that HD cooling is unimportant in cases where the primordial
gas does not go through an ionized phase, as in the formation process of the
very first stars in z ~ 20 minihaloes of mass ~ 10^{6} M_{solar}.
