  Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real
random variables with mean zero, variance one and with subexponential tail. We
show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More
precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf
{P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log
n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le
x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}
