With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs ${Gamma_n}_{ngeq 1}$, where $Gamma_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $partial T$ gives rise to orbital Schreier graphs $Gamma_{xi}$, $xiin partial T$. Denoting by $xi_n$ the prefix of length $n$ of the infinite ray $xi$, the rooted graph $(Gamma_{xi},xi)$ is then the limit of the sequence of finite rooted graphs ${(Gamma_n,xi_n)}_{ngeq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(Gamma_{xi},xi)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $xi$.