  We are just starting to understand the physical processes driving the
dramatic change in cosmic star-formation rate between $z\sim 2$ 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, thanks to the large redshift path surveyed ($\Delta z=182$),
we provide constraints on the molecular column density distribution function
beyond $z\sim 0$. In addition, we probe column densities of N(H$_2$) >
10$^{16}$ atoms~cm$^{-2}$, five orders of magnitude lower than in previous
studies. We use the cosmological hydrodynamical simulation IllustrisTNG to show
that our upper limits of $\rho ({\rm H}_2)\lesssim 10^{8.3} \text{M}_{\odot}
\text{Mpc}^{-3}$ at $0 < z \leq 1.7$ 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 $z\sim 2$.
