  We present ISO observations of several H2 pure-rotational lines (from S(0) to
S(5)) towards a sample of 16 molecular clouds distributed along the central ~
500 pc of the Galaxy. We also present C18O and 13CO J=1->0 and J=2->1
observations of these sources made with the IRAM-30m telescope. With the CO
data we derive H2 densities of 10e(3.5-4.0) cm-3 and H2 column densities of a
few 10e22 cm-2. We have corrected the H2 data for ~ 30 magnitudes of visual
extinction using a self-consistent method. In every source, we find that the H2
emission exhibits a large temperature gradient. The S(0) and S(1) lines trace
temperatures (T) of ~150 K while the S(4) and S(5) lines indicate temperatures
of ~ 600K. The warm H2 column density is typically ~1-2 x 10e22 cm-2, and is
predominantly gas with T=150 K. This is the first direct estimate of the total
column density of the warm molecular gas in the Galactic center region. These
warm H2 column densities represent a fraction of ~ 30 % of the gas traced by
the CO isotopes emission. The cooling by H2 in the warm component is comparable
to that by CO. Comparing our H2 and CO data with available ammonia NH3
observations from literature one obtains relatively high NH3 abundances of a
few 10e(-7) in both the warm and the cold gas. A single shock or
Photo-Dissociation Region (PDR) cannot explain all the observed H2 lines.
Alternatives for the heating mechanisms are discussed.
