We present the largest homogeneous survey of redshift > 4.4 damped Lyα systems (DLAs) using the spectra of 163 quasars that comprise the Giant Gemini GMOS (GGG) survey. With this survey we make the most precise high-redshift measurement of the cosmological mass density of neutral hydrogen, ΩHI. After correcting for systematic effects using a combination of mock and higher-resolution spectra, we find ΩHI= 0.98+0.20-0.18 × 10−3 at 〈z〉 = 4.9, assuming a 20% contribution from lower column density systems below the DLA threshold. By comparing to literature measurements at lower redshifts, we show that ΩHI can be described by the functional form ΩHI(z) ∝ (1 + z)0.4. This gradual decrease from z = 5 to 0 suggests that in the galaxies which dominate the cosmic star formation rate, Hi is a transitory gas phase fuelling star formation which must be continually replenished by more highly-ionized gas from the intergalactic medium, and from recycled galactic winds.