  (Abridged) We have used the Far Ultraviolet Spectroscopic Explorer (FUSE) to
search for H_2 absorption in five starburst galaxies: NGC 1705, NGC 3310, NGC
4214, M83 (NGC 5236), and NGC 5253. We tentatively detect weak absorption by
H_2 in M83 and NGC 5253, and set upper limits on the H_2 column density in the
other galaxies. Conservative upper limits on the mass of molecular gas detected
with FUSE are many orders of magnitude lower than the H_2 mass inferred from CO
emission. This indicates that almost all of the H_2 is in the form of clouds
with N(H_2)>10^20 cm^-2 that are opaque to far-UV light and therefore cannot be
probed with far-UV absorption measurements. The far-UV continuum visible in the
FUSE spectra passes between the dense clouds, which have a covering factor <1.
The complex observational biases related to varying extinction across the
extended UV emission in the FUSE apertures prevent an unambiguous
characterization of the diffuse H_2 in these starbursts. However, the evidence
is suggestive that there is less H_2 in the diffuse interstellar medium between
the dense clouds compared to similarly reddened sight lines in the Milky Way.
This holds with the expectation that the destructive UV radiation field is
stronger in starbursts. However, previous UV observations of these starbursts
have shown that there is reddening caused by the diffuse interstellar medium.
This suggests that while diffuse H_2 may be destroyed in the starburst, dust
still exists.
