  This paper takes a significant step towards confirming a long-standing and
far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They
conjectured in 1993 that there are no non-trivial block-transitive 6-designs.
We prove that the Cameron-Praeger conjecture is true for the important case of
non-trivial Steiner 6-designs, i.e. for 6-$(v,k,\lambda)$ designs with
$\lambda=1$, except possibly when the group is $P\GammaL(2,p^e)$ with $p=2$ or
3, and $e$ is an odd prime power.
