  Let $A$ be a separable $C^*$-algebra and $B$ a stable $C^*$-algebra
containing a strictly positive element. We show that the group $\Ext(SA,B)$ of
unitary equivalence classes of extensions of $SA$ by $B$, modulo the extensions
which are asymptotically split, coincides with the group of homotopy classes of
such extensions. This is done by proving that the Connes-Higson construction
gives rise to an isomorphism between $\Ext(SA,B)$ and the $E$-theory group
$E(A,B)$ of homotopy classes of asymptotic homomorphisms from $S^2A$ to $B$.
