In the book "Gravitation and cosmology" by Weinberg at the page 99-100. He defines the Levi-Civita tensor as $\epsilon^{0123}=+1$ from which he writes
$\epsilon_{\mu\nu\rho\sigma}=g_{\mu\mu'}g_{\nu\nu'}g_{\rho\rho'}g_{\sigma\sigma'} \epsilon^{\mu'\nu'\rho'\sigma'}$
But he concludes that it gives $\epsilon_{\mu\nu\rho\sigma}=-g \epsilon^{\mu\nu\rho\sigma}$
I don't understand the minus sign. If I consider the metric $g$ to be diagonal, we have
$\epsilon_{0123}=g_{00}g_{11}g_{22}g_{33} \epsilon^{0123}$ which gives $\epsilon_{0123}=g \epsilon^{0123}$
Is it a typo in his book or is there something I don't understand ?