We report the discovery of significant excess absorption in the X-ray spectra of 12 clusters of galaxies. The spectra, from the Einstein Observatory Solid State Spectrometer, also require a cooling-flow component, which confirms the results of imaging studies of the clusters showing the strongly peaked emission characteristic of cooling flows. Several other clusters, such as the Coma cluster, do not require either extra absorption of a cooling flow component. The total mass of absorbing gas is approximately 3 x 10(11)-10(12) M., on the assumption that it is distributed through the cooling flow region and has cosmic abundance. It represents a significant new component of the intracluster medium. We show that the gas is most likely in the form of small cold clouds. It may be difficult to detect at other wavelengths, although further studies of the 21-cm line of hydrogen should be most successful. The absorbing gas should be readily observable in spectra from the X-ray detectors on BBXRT and ROSAT.