  We show that the spectral radius of an $N\times N$ random symmetric matrix
with i.i.d. bounded centered but non-symmetrically distributed entries is
bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon}), $ where $\sigma^2 $
is the variance of the matrix entries and $\epsilon $ is an arbitrary small
positive number. Our bound improves the earlier results by Z.F\"{u}redi and
J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).
