  We derive total (atomic + molecular) hydrogen densities in giant molecular
clouds (GMCs) in the nearby spiral galaxy M33 using a method that views the
atomic hydrogen near regions of recent star formation as the product of
photodissociation. Far-UV photons emanating from a nearby OB association
produce a layer of atomic hydrogen on the surfaces of nearby GMCs. Our approach
provides an estimate of the total hydrogen density in these GMCs from
observations of the excess far-UV emission that reaches the GMC from the OB
association, and the excess 21-cm radio HI emission produced after these far-UV
photons convert H2 into HI on the GMC surface. The method provides an
alternative approach to the use of CO emission as a tracer of H2 in GMCs, and
is especially sensitive to a range of density well below the critical density
for CO(1-0) emission. We describe our "PDR method" in more detail and apply it
using GALEX far-UV and VLA 21-cm radio data to obtain volume densities in a
selection of GMCs in the nearby spiral galaxy M33. We have also examined the
sensitivity of the method to the linear resolution of the observations used;
the results obtained at 20 pc are similar to those for the larger set of data
at 80 pc resolution. The cloud densities we derive range from 1 to 500 cm-3,
with no clear dependence on galactocentric radius; these results are generally
similar to those obtained earlier in M81, M83, and M101 using the same method.
