  Building on the work of Nogin \cite{Nogin}, we prove that the braid group
$B_4$ acts transitively on full exceptional collections of vector bundles on
Fano threefolds with $b_2=1$ and $b_3=0$. Equivalently, this group acts
transitively on the set of simple helices (considered up to a shift in the
derived category) on such a Fano threefold. We also prove that on threefolds
with $b_2=1$ and very ample anticanonical class, every exceptional coherent
sheaf is locally free.
