  We study the minimal complex surfaces of general type with $p_g=0$ and
$K^2=7$ or 8 whose bicanonical map is not birational. In the paper 'The
bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$' we have shown that if
$S$ is such a surface, then the bicanonical map has degree 2. Here we describe
precisely such surfaces showing that there is a fibration $f\colon S\to \pp^1$
such that: i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve;
ii) the involution induced by the bicanonical map of $S$ restricts to the
hyperelliptic involution of $F$. Furthermore, if $K^2_S=8$, then $f$ is an
isotrivial fibration with 6 double fibres, and if $K^2_S=7$, then $f$ has 5
double fibres and it has precisely one fibre with reducible support, consisting
of two components.
