Molecular hydrogen (H2) is by far the most abundant material from which stars, protoplanetary disks and giant planets form, but it is difficult to detect directly. Infrared emission lines from H2 have recently been reported towards beta Pictoris, a star harbouring a young planetary system. This star is surrounded by a dusty 'debris disk' that is continuously replenished either by collisions between asteroidal objects or by evaporation of ices on Chiron-like objects. A gaseous disk has also been inferred from absorption lines in the stellar spectrum. Here we present the far-ultraviolet spectrum of beta Pictoris, in which H2 absorption lines are not seen. This allows us to set a very low upper limit on the column density of H2: N(H2) </= 1018 cm-2. This non-detection is puzzling when compared to the quantity of H2 inferred from the infrared observations, but it does show that H2 is not in the disk on the direct line of sight. Carbon monoxide (CO) has been seen in absorption against the star, yielding a ratio of CO/H2 > 6 x 10-4. As CO would be destroyed under ambient conditions in about 200 years (refs 9, 11), our result demonstrates that the CO in the disk arises from evaporation of planetesimals.