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.
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