  We prove an analog of Boettcher's theorem for transcendental entire functions
in the Eremenko-Lyubich class B. More precisely, let f and g be entire
functions with bounded sets of singular values and suppose that f and g belong
to the same parameter space (i.e., are *quasiconformally equivalent* in the
sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to
the set of points which remain in some sufficiently small neighborhood of
infinity under iteration. Furthermore, this conjugacy extends to a
quasiconformal self-map of the plane.
  We also prove that this conjugacy is essentially unique. In particular, we
show that an Eremenko-Lyubich class function f has no invariant line fields on
its escaping set.
  Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f
and g which belong to the same parameter space are conjugate on their sets of
escaping points.
