  We study the effects of WIMP dark matter (DM) annihilations on the thermal
and chemical evolution of the gaseous clouds where the first generation of
stars in the Universe is formed. We follow the collapse of the gas inside a
typical halo virializing at very high redshift, from well before virialization
until a stage where the heating from DM annihilations exceeds the gas cooling
rate. The DM energy input is estimated by inserting the energy released by DM
annihilations (as predicted by an adiabatic contraction of the original DM
profile) in a spherically symmetric radiative transfer scheme. In addition to
the heating effects of the energy absorbed, we include its feedback upon the
chemical properties of the gas, which is critical to determine the cooling rate
in the halo, and hence the fragmentation scale and Jeans mass of the first
stars. We find that DM annihilation does alter the free electron and especially
the H2 fraction when the gas density is n>~ 10^4 cm^-3, for our fiducial
parameter values. However, even if the change in the H2 abundance and the
cooling efficiency of the gas is large (sometimes exceeding a factor 100), the
effects on the temperature of the collapsing gas are far smaller (a reduction
by a factor <~1.5), since the gas cooling rate depends very strongly on
temperature: then, the fragmentation mass scale is reduced only slightly,
hinting towards no dramatic change in the initial mass function of the first
stars.
