Knowledge about the outermost portions of galaxies is limited owing to the small amount of light coming from them. It is known that in many cases atomic hydrogen (H I) extends well beyond the optical radius. In the centres of galaxies, however, molecular hydrogen (H2) usually dominates by a large factor, raising the question of whether H2 is also abundant in the outer regions. Here we report the detection of emission from carbon monoxide (CO), the most abundant tracer of H2, beyond the optical radius of the nearby galaxy NGC 4414. The host molecular clouds probably formed in the regions of relatively high H i column density and in the absence of spiral density waves. The relative strength of the lines from the two lowest rotational levels indicates that both the temperature and density of the H2 are quite low compared to conditions closer to the centre. The inferred surface density of the molecular material continues the monotonic decrease from the inner regions. We conclude that although molecular clouds can form in the outer region of this galaxy, there is little mass associated with them.