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A Dirac-type operator on a complete Riemannian manifold is of Callias-type ifits square is a Schr\"{o}dinger-type operator with a potential uniformlypositive outside of a compact set. We develop the theory of Callias-typeoperators twisted with Hilbert $C^\ast$-module bundles and prove an indextheorem for such operators. As an application, we derive an obstruction to theexistence of complete Riemannian metrics of positive scalar curvature onnoncompact spin manifolds in terms of closed submanifolds of codimension-one.

Callias-type operators in $C^\ast$-algebras and positive scalar curvature on noncompact manifolds