  The weak decays of a spin-$1/2$ doubly charm baryon (${\cal B}_{cc}$) to a
spin-$1/2$ singly charm baryon (${\cal B}_c$) and a light vector meson ($V$)
are studied under a phenomenological scheme. The contributions are classified
into different topological diagrams, among which the short distance ones are
calculated under the factorization hypothesis, and the long distance
contributions are modelled as final-state interactions (FSIs) which are
estimated with the one-particle-exchange model. In calculation the topological
contributions tend to fall in a hierarchy. The branching fractions or decay
widths are estimated, and it indicates that $\Xi_{cc}^+\to\Xi_c^+\pi^+\pi^-$
and $\Omega_{cc}^{+}\rightarrow\Xi_{c}^{+}K^-\pi^+$ can be used as candidate
decays for searching $\Xi_{cc}^+$ and $\Omega_{cc}^+$. Some decays that are
mainly activated by the long distance effects are found, observation on which
in future experiments can help to understand the role of FSIs in charm baryon
decays.
