It is shown that the Elliott invariant is a complete invariant for the simple unital $C^*$-algebras which can be realized as an inductive limit of a sequence of finite direct sums of algebras of the form $$ \left \{f \in C({\mathbb T}) \otimes M_n\mid f(x_i) \in M_d, i = 1,2, \ldots , N \right \},$$ where $(x_1,x_2, \ldots , x_N)$ is an arbitrary (finite) set on the circle $\mathbb T$ and $d$ is a natural number dividing $n$. The corresponding range of invariants is identified and the classification result is extended to the non-unital case. A series of results about the structure of these $C^*$-algebras and the maps between them are also obtained.