We are just starting to understand the physical processes driving the dramatic change in cosmic star formation rate between z ∼ 2 and the present day. A quantity directly linked to star formation is the molecular gas density, which should be measured through independent methods to explore variations due to cosmic variance and systematic uncertainties. We use intervening CO absorption lines in the spectra of mm-bright background sources to provide a census of the molecular gas mass density of the Universe. The data used in this work are taken from ALMACAL, a wide and deep survey utilizing the ALMA calibrator archive. While we report multiple Galactic absorption lines and one intrinsic absorber, no extragalactic intervening molecular absorbers are detected. However, due to the large redshift path surveyed (∆z = 182), we provide constraints on the molecular column density distribution function beyond z ∼ 0. In addition, we probe column densities of N(H₂) > 10¹⁶ atoms cm^-2, 5 orders of magnitude lower than in previous studies. We use the cosmological hydrodynamical simulation IllustrisTNG to show that our upper limits of ρ (H_2)≲ 10^{8.3} M_{☉ } Mpc^{-3} at 0 < z ≤ 1.7 already provide new constraints on current theoretical predictions of the cold molecular phase of the gas. These results are in agreement with recent CO emission-line surveys and are complementary to those studies. The combined constraints indicate that the present decrease of the cosmic star formation rate history is consistent with an increasing depletion of molecular gas in galaxies compared to z ∼ 2....