  In this paper we study ensembles of random symmetric matrices $\X_n =
{X_{ij}}_{i,j = 1}^n$ with dependent entries such that $\E X_{ij} = 0$, $\E
X_{ij}^2 = \sigma_{ij}^2$, where $\sigma_{ij}$ may be different numbers.
Assuming that the average of the normalized sums of variances in each row
converges to one and Lindeberg condition holds we prove that the empirical
spectral distribution of eigenvalues converges to Wigner's semicircle law.
