  In an old paper of the author the thinnest five-neighbour packing of
translates of a convex disc (different from a parallelogram) was determined.
The minimal density was $3/7$, and was attained for a certain packing of
triangles. In that paper it was announced that for centrally symmetric convex
plates (different from a parallelogram) the analogous minimal density was
$9/14$, and was attained for a certain packing of affine regular hexagons, and
a very sketchy idea of the proof was given. In this paper we give details of
this proof.
