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Let F be an arbitrary local field. Consider the standard embedding of GL(n,F)into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F).  In this paper we show that any GL(n,F) \times GL(n,F)-invariant distributionon GL(n+1,F) is invariant with respect to transposition.  We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfandpair. Namely, for any irreducible admissible representation $(\pi,E)$ of(GL(n+1,F), $$dimHom_{GL(n,F)}(E,\cc) \leq 1.$$  For the proof in the archimedean case we develop several new tools to studyinvariant distributions on smooth manifolds.

(GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F