  Context At least a fraction of the atomic hydrogen in spiral galaxies is
suspected to be the result of molecular hydrogen which has been dissociated by
radiation from massive stars.
  Aims In this paper, we extend our earlier set of data from a small region of
the Western spiral arm of M81 with CO observations in order to study the
interplay between the radiation field and the molecular and atomic hydrogen.
  Methods We report CO(1-0) observations with the Nobeyama 45 m dish and the
Owens Valley interferometer array of selected regions in the Western spiral arm
of M81.
  Results From our Nobeyama data, we detect CO(1-0) emission at several
locations, coinciding spatially with HI features near a far-UV source. The
levels and widths of the detected CO profiles are consistent with the CO(1-0)
emission that can be expected from several large photo-dissociation regions
with typical sizes of some 50x150 pc located within our telescope beam. We do
not detect emission at other pointings, even though several of those are near
far-UV sources and accompanied by bright HI. This non-detection is likely a
consequence of the marginal area filling factor of photo-dissociation regions
in our observations. We detect no emission in our Owens Valley data, consistent
with the low intensity of the CO emission detected in that field by the
Nobeyama dish.
  Conclusions We explain the lack of CO(1-0) emission at positions farther from
far-UV sources as a consequence of insufficient heating and excitation of the
molecular gas at these positions, rather than as an absence of molecular
hydrogen.
