The well-known Steinhaus theorem (2) with respect to the set of distances of linear sets of positive Lebesgue measure has been generalized to the case of linearly measurable subsets of rectifiable curves in the Euclidean plane by Besicovitch and Miller (1). An extension of the theorem to rectifiable curves in Euclidean n-space is immediate. Prof. A. P. Morse has suggested the problem as to whether or not the theorem still remains true in a general metric space. By defining a particular curve ℒ in a metric space we prove that the answer to this question is in the negative.