Some X-ray observations of cooling-flow clusters show soft X-ray absor
ption exceeding that expected along the line of sight through our own
Galaxy. This absorption appears at the position of the cooling flow an
d covers a similar solid angle at the center of the cluster. The infer
red absorbing column densities correspond to a hydrogen mass exceeding
10(11) M., prompting suggestions that the absorbing material is conde
nsed gas accumulated from the cooling flow. We explore the characteris
tics of cold atomic clouds embedded in an X-ray-emitting cooling flow
and find that, if they cover the central 100 kpc of the cluster, they
should already have been detected in H I 21 cm emission. Dust in cooli
ng-flow clouds can catalyze molecule formation, making them unobservab
le at 21 cm, but dusty molecular clouds should radiate detectable, opt
ically thick CO rotational lines, which likewise have not been seen. X
-ray transient heating of grains prohibits most of the CO from condens
ing onto grain surfaces and thus ensures that the CO lines are optical
ly thick. Ionized X-ray-absorbing gas would radiate profusely in optic
al, UV, or X-ray emission lines. We report limits on Her and [Fe x] 63
74 Angstrom surface brightnesses from deep long-slit spectroscopy that
rule out ionized columns thicker than 10(21) cm(-2) and cooler than 1
.5 x 10(6) K. Limits on O VIII Ly alpha do not allow the X-ray-absorbi
ng gas to be at higher temperatures. One remaining possibility is that
dust in the hot intracluster medium absorbs the soft X-rays. The soft
X-ray opacity of dust is similar to its optical opacity. Optical exti
nctions inferred from the deficits of background galaxies and quasars
counted behind clusters might be consistent with the dust column densi
ties inferred from soft X-ray absorption. If dust is the culprit, limi
ts on the 100 mu m luminosities of clusters imply that the dust-to-gas
ratio must be higher at similar to 1 Mpc, at which large grains can s
urvive for longer than 10(9) yr, than in the cores of clusters, where
sputtering destroys grains on a much shorter timescale. However, dust
at similar to 1 Mpc in quantities sufficient to produce significant so
ft X-ray absorption represents a large fraction of the total metal con
tent of a cluster. Submillimeter continuum observations should eventua
lly determine whether dust is widespread in the intracluster media of
clusters of galaxies.