  Let $P$ be a point of the Veronese surface $\Vcal$ in \PG53. Then thereare
four conics of $\Vcal$ through $P$. We show that the internal points of those
conics form a 12-cap which is a point model for Witt's 5-$(12,6,1)$ design. In
fact, this construction is "dual" to a similar construction that has been
established by the author. We give an explicit parametrization of the cap
$\Kcal$; the domain is a dual affine plane which arises from \PG23 by removing
one point. Thus, as a by--product, we obtain an easy approach to the extended
ternary Golay code $G_{12}$. Finally, we discuss some other procedures that
yield 12-sets of points from the Veronese surface.
