Singular spaces are playing an increasingly important role in Riemannian geometry. In particular this is true of the so-called Alexandrov spaces; i.e., finite-(Hausdorff) dimensional, complete, inner metric spaces with a lower curvature bound in the local triangle comparison sense. Interest in Alexandrov spaces is largely explained by the fact that the natural process of taking Gromov-Hausdorff limits is closed in Alexandrov geometry but not in Riemannian geometry. This simple observation has become a powerful tool in Riemannian geometry (see, for example, [FY], [GP2], or [GPW] and [Pe]). In this paper, we use a new means to obtain results in Riemannian geometry by studying Alexandrov spaces. The operation of forming spherical suspensions of positively curved spaces, an operation which is also closed in Alexandrov geometry but not in Riemannian geometry, will be applied to prove a differentiable sphere theorem which is unlike all previous solutions to the following basic problem.