  Using spatially resolved, deprojected ROSAT PSPC spectra of 10 of the
brightest cooling flow galaxies and groups with low Galactic column densities
we have detected intrinsic absorption over energies ~0.4-0.8 keV in half of the
sample. Since no intrinsic absorption is indicated for energies below ~0.4 keV,
the most reasonable model for the absorber is collisionally ionized gas at
temperatures T=10^{5-6} K with most of the absorption arising from ionized
states of oxygen but with a significant contribution from carbon and nitrogen.
The soft X-ray emission of this warm gas can explain the sub-Galactic column
densities of cold gas inferred within the central regions of most of the
systems. Attributing the absorption to ionized gas reconciles the large columns
of cold H and He inferred from EINSTEIN and ASCA with the lack of such columns
inferred from ROSAT. Within the central ~10-20 kpc, where the constraints are
most secure, the estimated mass of the ionized absorber is consistent with most
(perhaps all) of the matter deposited by a cooling flow over the lifetime of
the flow. Since the warm absorber produces no significant H or He absorption
the large absorber masses are consistent with the negligible atomic and
molecular H inferred from HI and CO observations of cooling flows. It is also
found that if T > ~2x10^5 K then the optical and UV emission implied by the
warm gas does not violate published constraints. Finally, we discuss how the
prediction of warm ionized gas as the product of mass drop-out in these and
other cooling flows can be verified with new CHANDRA and XMM observations.
(Abridged)
