In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending
structures problem for associative conformal algebras. Using the unified
product as a tool, which includes interesting products such as bicrossed
product, cocycle semi-direct product and so on, a cohomological type object is
constructed to characterize the $\mathbb{C}[\partial]$-split extending
structures for associative conformal algebras. Moreover, using this theory, the
extending structures of an associative conformal algebra $A$ which is free as a
$\mathbb{C}[\partial]$-module by the $\mathbb{C}[\partial]$-module
$Q=\mathbb{C}[\partial]x$ are described using flag datums of $A$. Furthermore,
we give a classification of the extending structures of $A$ by
$Q=\mathbb{C}[\partial]x$ in detail up to equivalence when $A$ is a free
associative conformal algebra of rank 1.Comment: 17 pages, Linear and Multillinear Algebra, 201