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Given a metric continuum X, we consider the hyperspace Cn(X) of all nonempty closed subsets of X with at most n components. In this paper we prove that if n 6= 2, X is an indecomposable continuum such that all its proper nondegenerate subcontinua are arcs and Y is a continuum such that Cn(X) is homeomorphic to Cn(Y ), then X is homeomorphic to Y (that is, X has unique hyperspace Cn(X)). Subject Mathematics Classication: Primary 54B20, Secondary: 54F15

Uniqueness of hyperspaces of indecomposable arc continua