  We give a short proof of the following result: Let $X$ be a complex surface
of general type. If the canonical divisor of the minimal model of $X$ has
selfintersection $= 1$, then $X$ is not diffeomorphic to a rational surface.
Our proof is the natural extension of the argument given in our paper in
Inventiones Mathematicae (1989) for the case when $X$ is minimal. This argument
also gives information about the non--existence of certain smooth embeddings of
$2$--spheres in $X$, if $X$ has geometric genus zero.
