  Despite its crucial role in galaxy evolution, the complex circumgalactic
medium (CGM) remains underexplored. Although it is known to be multi-phase, the
importance of the molecular gas phase to the total CGM mass budget is, to date,
unconstrained. We present the first constraints on the molecular gas covering
fraction in the CGM of low-redshift galaxies, using measurements of CO column
densities along sightlines towards mm-bright background quasars with
intervening galaxies. We do not detect molecular absorption against the
background quasars. For the individual, low-redshift, 'normal' galaxy haloes
probed here, we can therefore rule out the presence of an extremely molecular
gas-rich CGM, as recently reported in high-redshift protoclusters and around
luminous active galactic nuclei. We also set statistical limits on the volume
filling factor of molecular material in the CGM as a whole, and as a function
of radius. ISM-like molecular clouds of ~30 pc in radius with column densities
of N(CO) >~ 10^16 cm^-2 have volume filling factors of less than 0.2 per cent.
Large-scale smooth gas reservoirs are ruled out much more stringently. The
development of this technique in the future will allow deeper constraining
limits to be set on the importance (or unimportance) of molecular gas in the
CGM.
