  We explore the effects of magnetic energy dissipation on the formation of the
first stars. For this purpose, we follow the evolution of primordial chemistry
in the presence of magnetic fields in the post-recombination universe until the
formation of the first virialized halos. From the point of virialization, we
follow the protostellar collapse up to densities of $\sim10^{12}$ cm$^{-3}$ in
a one-zone model. In the intergalactic medium (IGM), comoving field strengths
of $\gtrsim0.1$ nG lead to Jeans masses of $10^8 M_\odot$ or more and thus
delay gravitational collapse in the first halos until they are sufficiently
massive. During protostellar collapse, we find that the temperature minimum at
densities of $\sim10^3$ cm$^{-3}$ does not change significantly, such that the
characteristic mass scale for fragmentation is not affected. However, we find a
significant temperature increase at higher densities for comoving field
strengths of $\gtrsim0.1$ nG. This may delay gravitational collapse, in
particular at densities of $\sim10^9$ cm$^{-3}$, where the proton abundance
drops rapidly and the main contribution to the ambipolar diffusion resistivity
is due to collisions with Li$^+$. After the formation of the protostar, the
increased gas temperatures may enhance the protostellar accretion rate. Our
model confirms that initial weak magnetic fields may be amplified considerably
during gravitational collapse and become dynamically relevant. For instance, a
comoving field strength above $10^{-5}$ nG will be amplified above the critical
value for the onset of jets which can magnetize the IGM.
