Convergence of formal embeddings between real-analytic hypersurfaces in codimension one

We show that every formal embedding sending a real-analytic strongly pseudoconvex hypersurface in $M\subset {\text{C}}^N$ into another such hypersurface in $M^{\mathrm{\prime }}\subset {\text{C}}^{N+1}$ is convergent. More generally, if $M$ and $M^{\mathrm{\prime }}$ are merely Levi-nondegenerate, the same conclusion holds for any formal embedding provided either that the embedding is CR transversal or the target hypersurface does not contain any complex curves.