  Let $S$ be a minimal surface of general type with $p_g=0$ and $K^2=6$, such
that its bicanonical map $\fie\colon S\to\pp^6$ is not birational. The map
$\fie$ is a morphism of degree $\le 4$ onto a surface. The case of $\deg\fie=4$
is completely characterized in [Topology, {\bf 40} (5)
  (2001), 977--991] and the present paper completes the classification of these
surfaces. It is proven that the degree of $\fie$ cannot be equal to 3, and the
geometry of surfaces with $\deg\fie=2$ is analysed in detail. The last section
contains three examples of such surfaces, two of which appear to be new.
