  We investigate which free constructions (amalgamated products and
HNN-extensions) over word hyperbolic groups produce groups that are again word
hyperbolic. A complete answer is obtained for the case when the amalgamated
subgroups are virtually cyclic. The results are applied, in particular, to show
that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable
word problem and conjugacy problem.
