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Abstract We report that the neutral hydrogen (Hi) mass density of the Universe (ρ_Hi) increases with cosmic time sincez ∼ 5, peaks atz ∼ 3, and then decreases towardz ∼ 0. This is the first result of Qz5, our spectroscopic survey of 63 quasars atz ≳ 5 with VLT/X-SHOOTER and Keck/ESI aimed at characterizing intervening Higas absorbers atz ∼ 5. The main feature of Qz5 is the high resolution (R ∼ 7000–9000) of the spectra, which allows us to (1) accurately detect high column density Higas absorbers in an increasingly neutral intergalactic medium atz ∼ 5 and (2) determine the reliability of previousρ_Himeasurements derived with lower resolution spectroscopy. We find five intervening damped Lyαabsorbers (DLAs) atz > 4.5, which corresponds to the lowest DLA incidence rate ( $0.034_{0.02}^{0.05}$) atz ≳ 2. We also measure the lowestρ_Hiatz ≳ 2 from our sample of DLAs and subDLAs, corresponding toρ_Hi $=0.56_{0.31}^{0.82}\times 10^8{M}_{\odot }$Mpc⁻³atz ∼ 5. Taking into account our measurements atz ∼ 5 and systematic biases in the DLA detection rate at lower spectral resolutions, we conclude thatρ_Hidoubles fromz ∼ 5 toz ∼ 3. From these results emerges a qualitative agreement between how the cosmic densities of Higas mass, molecular gas mass, and star formation rate build up with cosmic time.