  We present the largest homogeneous survey of $z>4.4$ damped Lyman-$\alpha$
systems (DLAs) using the spectra of 163 QSOs 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, $\Omega_{\rm
HI}$. At such high redshift important systematic uncertainties in the
identification of DLAs are produced by strong intergalactic medium absorption
and QSO continuum placement. These can cause spurious DLA detections, result in
real DLAs being missed, or bias the inferred DLA column density distribution.
We correct for these effects using a combination of mock and higher-resolution
spectra, and show that for the GGG DLA sample the uncertainties introduced are
smaller than the statistical errors on $\Omega_{\rm HI}$. We find $\Omega_{\rm
HI}=0.98^{+0.20}_{-0.18}\times10^{-3}$ at $\langle z\rangle=4.9$, assuming a
20% contribution from lower column density systems below the DLA threshold. By
comparing to literature measurements at lower redshifts, we show that
$\Omega_{\rm HI}$ can be described by the functional form $\Omega_{\rm
HI}(z)\propto(1+z)^{0.4}$. This gradual decrease from $z=5$ to $0$ is
consistent with the bulk of HI gas being a transitory phase fuelling star
formation, which is continually replenished by more highly-ionized gas from the
intergalactic medium, and from recycled galactic winds.
