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We report that the neutral hydrogen (H i) mass density of the Universe (ρH i) increases with cosmic time since z ∼ 5, peaks at z ∼ 3, and then decreases toward z ∼ 0. This is the first result of Qz5, our spectroscopic survey of 63 quasars at z ≳ 5 with VLT/X-SHOOTER and Keck/ESI aimed at characterizing intervening H i gas absorbers at z ∼ 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 H i gas absorbers in an increasingly neutral intergalactic medium at z ∼ 5 and (2) determine the reliability of previous ρH i measurements derived with lower resolution spectroscopy. We find five intervening damped Lyα absorbers (DLAs) at z > 4.5, which corresponds to the lowest DLA incidence rate () at z ≳ 2. We also measure the lowest ρH i at z ≳ 2 from our sample of DLAs and subDLAs, corresponding to ρH i Mpc−3 at z ∼ 5. Taking into account our measurements at z ∼ 5 and systematic biases in the DLA detection rate at lower spectral resolutions, we conclude that ρH i doubles from z ∼ 5 to z ∼ 3. From these results emerges a qualitative agreement between how the cosmic densities of H i gas mass, molecular gas mass, and star formation rate build up with cosmic time.