  We show that X-ray clusters would have cooled substantially over a Hubble
time by transport of heat from their hot interior to the their envelope, if the
heat conductivity had not been heavily suppressed relative to the Spitzer value
due to magnetic fields. The suppression is required in order for the observed
abundance of hot X-ray clusters to be consistent with predictions from popular
cosmological models. If a similar or stronger suppression factor applies to
cluster cores, then thermal conduction can not be the mechanism that prevents
cooling flows there.
