  The aim of this paper is to provide an answer to the
$\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal
algebras, which asks that how to describe all Leibniz conformal algebra
structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz
conformal subalgebra. For this purpose, an unified product of Leibniz conformal
algebras is introduced. Using this tool, two cohomological type objects are
constructed to classify all such extending structures up to an isomorphism.
Then this general theory is applied to the special case when $R$ is a free
$\mathbb{C}[\partial]$-module and $Q$ is a free $\mathbb{C}[\partial]$-module
of rank one. Finally, the twisted product, crossed product and bicrossed
product between two Leibniz conformal algebras are introduced as special cases
of the unified product, and some examples are given.
