  We know little about the outermost portions of galaxies because there is
little light coming from them. We do know that in many cases atomic hydrogen
(HI) extends well beyond the optical radius \cite{Casertano91}. In the centers
of galaxies, however, molecular hydrogen (H2) usually dominates by a large
factor, raising the question of whether H2 is abundant also in the outer
regions but hitherto unseen.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 molecular clouds probably formed in the
regions of relatively high HI 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 center. The inferred surface
density of the molecular material continues the monotonic decrease from the
inner regions. We conclude that while molecular clouds can form in the outer
region of this galaxy, there is little mass associated with them.
