  Recent studies of nearby spiral galaxies suggest that photodissociation
regions (PDRs) are capable of producing much of the observed HI in galaxy
disks. In that case, measurements of the HI column density and the
far-ultraviolet (FUV) photon flux provide a new probe of the volume density of
the local underlying H_2. We develop the method and apply it to the giant Scd
spiral M101 (NGC 5457). We find that, after correction for the best-estimate
gradient of metallicity in the ISM of M101 and for the extinction of the
ultraviolet emission, molecular gas with a narrow range of density from 30-1000
cm^-3 is found near star- forming regions at all radii in the disk of M101 out
to a distance of 12' (approximately 26 kpc), close to the photometric limit of
R_25 = 13.5'.
  In this picture, the ISM is virtually all molecular in the inner parts of
M101. The strong decrease of the HI column density in the inner disk of the
galaxy at R_G < 10 kpc is a consequence of a strong increase in the dust-to-gas
ratio there, resulting in an increase of the H_2 formation rate on grains and a
corresponding disappearance of hydrogen in its atomic form.
