Condition of conjugacy of $WCG$-spaces

It is proved that a $WCG$-space $E$ is conjugate to a Banach space if and only if its conjugate space $E'$ contains a norm-closed total subspace $M$, consisting of functionals which attain supremum on the unit sphere. Moreover, $M'=E$ in the duality established between $E$ and Eprime. An example, showing that this statement is in general not true for an arbitrary Banach space, is given.