  In this paper we study bielliptic curves of genus 3 defined over an
algebraically closed field $k$ and the intersection of the moduli space
$\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection
$\S$ is an irreducible, 3-dimensional, rational algebraic variety. We determine
the equation of this space in terms of the $Gl(2, k)$-invariants of binary
octavics as defined in \cite{hyp_mod_3} and find a birational parametrization
of $\S$. We also compute all possible subloci of curves for all possible
automorphism group $G$. Moreover, for every rational moduli point $\p \in \S$,
such that $| \Aut (\p) | > 4$, we give explicitly a rational model of the
corresponding curve over its field of moduli in terms of the $Gl(2,
k)$-invariants.

Algebraic Geometry