  In this paper, we study double structures supported on rational normal
curves. After recalling the general construction of double structures supported
on a smooth curve described in \cite{fer}, we specialize it to double
structures on rational normal curves. To every double structure we associate a
triple of integers $ (2r,g,n) $ where $ r $ is the degree of the support, $ n
\geq r $ is the dimension of the projective space containing the double curve,
and $ g $ is the arithmetic genus of the double curve. We compute also some
numerical invariants of the constructed curves, and we show that the family of
double structures with a given triple $ (2r,g,n) $ is irreducible. Furthermore,
we prove that the general double curve in the families associated to $
(2r,r+1,r) $ and $ (2r,1,2r-1) $ is arithmetically Gorenstein. Finally, we
prove that the closure of the locus containing double conics of genus $ g \leq
-2 $ form an irreducible component of the corresponding Hilbert scheme, and
that the general double conic is a smooth point of that component. Moreover, we
prove that the general double conic in $ \mathbb{P}^3 $ of arbitrary genus is a
smooth point of the corresponding Hilbert scheme.
