Context. Almost 200 different species have been detected in the interstellar medium (ISM) during the last decades, revealing not only simple species but complex molecules with more than 6 atoms. Other exotic compounds, like the weakly-bound dimer (H 2 ) 2 , have also been detected in astronomical sources like Jupiter. Aims. We aim at detecting for the first time the CO-H 2 van der Waals complex in the ISM, which if detected can be a sensitive indicator for low temperatures. Methods. We use the IRAM 30m telescope, located in Pico Veleta (Spain), to search for the CO-H 2 complex in a cold, dense core in TMC-1C (with a temperature of ∼ 10 K). All the brightest CO-H 2 transitions in the 3 mm (80-110 GHz) band have been observed with a spectral resolution of 0.5-0.7 km s −1 , reaching a rms noise level of ∼ 2 mK. The simultaneous observation of a broad frequency band, 16 GHz, has allowed us to conduct a serendipitous spectral line survey. Results. No lines belonging to the CO-H 2 complex have been detected. We have set up a new, more stringent upper limit for its abundance to be [CO-H 2 ]/[CO] ∼ 5 × 10 −6 , while we expect the abundance of the complex to be in the range ∼ 10 −8 -10 −3 . The spectral line survey has allowed us to detect 75 lines associated with 41 different species (including isotopologues). We detect a number of complex organic species, e.g. methyl cyanide (CH 3 CN), methanol (CH 3 OH), propyne (CH 3 CCH) and ketene (CH 2 CO), associated with cold gas (excitation temperatures ∼ 7 K), confirming the presence of these complex species not only in warm objects but also in cold regimes.