Let F be an arbitrary local field. Consider the standard embedding
and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F),
. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.