Cosmological hydrodynamical simulations of primordial star formation suggest that the gas within the first star-forming halos is turbulent. This has strong implications on the subsequent evolution, in particular on the generation of magnetic fields. Using high-resolution numerical simulations, we show that in the presence of turbulence, weak seed magnetic fields are exponentially amplified by the small-scale dynamo during the formation of the first stars. We conclude that strong magnetic fields are generated during the birth of the first stars in the universe, potentially modifying the mass distribution of these stars and influencing the subsequent cosmic evolution. We find that the presence of the small-scale turbulent dynamo can only be identified in numerical simulations in which the turbulent motions in the central core are resolved with at least 32 grid cells.