  Measurements of molecular hydrogen (H_2) column densities are presented for
the first six rotational levels (J=0 to 5) for 73 extragalactic targets
observed with FUSE. All of these have a final signal-to-noise ratio larger than
\snlimit, and are located at galactic latitude |b|>20 deg. The individual
observations were calibrated with the FUSE calibration pipeline CalFUSE version
2.1 or higher, and then carefully aligned in velocity. The final velocity
shifts for all the FUSE segments are listed. H_2 column densities or limits are
determined for the 6 lowest rotational (J) levels for each HI component in the
line of sight, using a curve-of-growth approach at low column densities ~16.5),
and Voigt-profile fitting at higher column densities. Detections include 73
measurements of low-velocity H_2 in the Galactic Disk and lower Halo. Eight
sightlines yield non-detections for Galactic H_2. The measured column densities
range from log N(H_2)=14 to log N(H_2)=20. Strong correlations are found
between log N(H_2) and T_01, the excitation temperature of the H_2, as well as
between log N(H_2) and the level population ratios (log (N(J')/N(J))). The
average fraction of nuclei in molecular hydrogen (f(H_2)) in each sightline is
calculated; however, because there are many HI clouds in each sightline, the
physics of the transition from HI to H_2 can not be studied. Detections also
include H2 in 16 intermediate-velocity clouds in the Galactic Halo (out of 35
IVCs). Molecular hydrogen is seen in one high-velocity cloud (the Leading Arm
of the Magellanic Stream), although 19 high-velocity clouds are intersected;
this strongly suggests that dust is rare or absent in these objects. Finally,
there are five detections of H_2 in external galaxies.
