The plasma in galaxy clusters is subject to firehose and mirror instabilities at scales of order the ion Larmor radius .
The mirror instability generates fluctuations of magnetic-field strength \delta B / B \sim 1 .
These fluctuations act as magnetic traps for the heat-conducting electrons , suppressing their transport .
We calculate the effective parallel thermal conductivity in the ICM in the presence of the mirror fluctuations for different stages of the evolution of the instability .
The mirror fluctuations are limited in amplitude by the maximum and minimum values of the field strength , with no large deviations from the mean value .
This key property leads to a finite suppression of thermal conduction at large scales .
We find suppression down to \approx 0.2 of the Spitzer value for the secular phase of the perturbations ’ growth , and \approx 0.3 for their saturated phase .
The effect operates in addition to other suppression mechanisms and independently of them .
Globally , fluctuations \delta B / B \sim 1 can be present on much larger scales , of the order of the scale of turbulent motions .
However , we do not expect large suppression of thermal conduction by these , because their scale is considerably larger than the collisional mean free path of the ICM electrons .
The obtained suppression of thermal conduction by a factor of \sim 5 appears to be characteristic and potentially universal for a weakly collisional mirror-unstable plasma .