  We use the present observed number density of large X-ray clusters to
constrain the amplitude of matter density perturbations induced by cosmic
strings on the scale of $8 h^{-1}$Mpc ($\sigma_8$), in both open cosmologies
and flat models with a non-zero cosmological constant. We find a slightly lower
value of $\sigma_8$ than that obtained in the context of primordial Gaussian
fluctuations generated during inflation. This lower normalization of $\sigma_8$
results from the mild non-Gaussianity on cluster scales, where the one point
probability distribution function is well approximated by a $\chi^2$
distribution. We use our estimate of $\sigma_8$ to constrain the string linear
energy density $\mu$ and show that it is consistent with the COBE
normalization.
