  Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface
$Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map
associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves
contained in $X$ which are preimages of lines bitangent to $Y$, gives an
isogeny between the Albanese variety of $\tilde F(X)$ and the intermediate
Jacobian of $X$.
