In order to constrain the initial mass function of the first generation of stars (Population III), we investigate the fragmentation properties of metal-free gas in the context of a hierarchical model of structure formation. We investigate the evolution of an isolated 3 sigma peak of mass 2x106 M middle dot in circle that collapses at zcoll approximately 30 using smoothed particle hydrodynamics. We find that the gas dissipatively settles into a rotationally supported disk that has a very filamentary morphology. The gas in these filaments is Jeans unstable with MJ approximately 103 M middle dot in circle. Fragmentation leads to the formation of high-density (n>108 cm-3) clumps that subsequently grow in mass by accreting the surrounding gas and by merging with other clumps up to masses of approximately 104 M middle dot in circle. This suggests that the very first stars were rather massive. We explore the complex dynamics of the merging and tidal disruption of these clumps by following their evolution over a few dynamical times.