We give a complete classification (upto unitary equivalence) of extensions of $C(X)$ by a separable simple AF-algebra $A$ with a unique trace (upto scalar multiples), where $X$ is a compact subset of the plane. In particular, we show that there arenon-trivial extensions $\tau$ such that $[\tau]=0$ in ${\rm Ext}(C(X),A).$ A new index is introduced todetermine when an extension is trivial. Extensions of $C(S^2)$ and other algebras are also studied. Ourresults work for a larger class of C*-algebras of real rank zero.

1991 Mathematics SubjectClassification: primary 46L05,46L35; secondary 46L80.