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The authors announce the following theorem.  Theorem 1. If $G=A*_H B$ is an amalgamated product where $A$ and $B$ arefinitely presented and semistable at infinity, and $H$ is finitely generated,then $G$ is semistable at infinity. If $G=A*_H$ is an HNN-extension where $A$is finitely presented and semistable at infinity, and $H$ is finitelygenerated, then $G$ is semistable at infinity.

Semistability of amalgamated products, HNN-extensions, and all one-relator groups