  We study the $4$-rank of the ideal class group of $K_n :=
\mathbb{Q}(\sqrt{-n}, \sqrt{n})$. Our main result is that for a positive
proportion of the squarefree integers $n$ we have that the $4$-rank of
$\text{Cl}(K_n)$ equals $\omega_3(n) - 1$, where $\omega_3(n)$ is the number of
prime divisors of $n$ that are $3$ modulo $4$.
