The first epoch of galaxy formation is governed by the infall of neutral, pristine gas. These neutral atomic hydrogen (HI) gas reservoirs will subsequently cool and condense into molecular clouds and initiate star-formation. The HI gas content is therefore a key ingredient in the overall process of galaxy evolution. In the local Universe the hyperfine HI 21-cm transition has been used as the main tracer of this neutral atomic gas, but due to the weakness of the line this approach is only feasible at moderate lookback distances for individual galaxies, even with next generation radio observatories. In this talk I will present a new approach to infer the HI gas mass of high-redshift galaxies, based on an empirical measurement of the [CII]-to-HI conversion factor using gamma-ray bursts. These bright cosmic beacons are used to illuminate the column density ratio of HI and [CII], which provides a scaling between the HI mass and [CII] luminosity per unit column in the line of sight. I will demonstrate how this conversion factor can be applied to recent galaxy samples surveying [CII] out to the edge of the epoch of reionization, at z~6. The HI gas mass is found to exceed the stellar mass at redshifts greater than z~1, and to increase as a function of redshift. Similarly, the fraction of HI to the total baryonic mass of these galaxies is observed to increases from around 25% at z=0 to about 60% at z~6. Further, I will show how the association of [CII] with HI also naturally explains the observed, more extended [CII] emission maps of high-redshift galaxies. I will also demonstrate how this technique makes it possible to infer the cosmic HI gas mass density in galaxies from z~6 to the present, based on estimates of the [CII] luminosity density. These results show the baryonic matter of star-forming galaxies in the early Universe is dominated by neutral atomic gas, a vital component to take into account when determining the gas available to initiate and maintain star formation.