  The singular part of Borel transform of a QCD amplitude near the infrared
renormalon can be expanded in terms of higher order Wilson coefficients of the
operators associated with the renormalon. In this paper we observe that this
expansion gives nontrivial constraints on the Borel amplitude that can be used
to improve the accuracy of the ordinary perturbative expansion of the Borel
amplitude. In particular, we consider the Borel transform of the Adler function
and its expansion around the first infrared renormalon due to the gluon
condensate. Using the next-to-leading order Wilson coefficient of the gluon
condensate operator, we obtain an exact constraint on the Borel amplitude at
the first IR renormalon. We then extrapolate, using judiciously chosen
conformal transformations and Pade approximants, the ordinary perturbative
expansion of the Borel amplitude in such a way that this constraint is
satisfied. This procedure allows us to predict the $O(\alpha_s^4)$ coefficient
of the Adler function, which gives a result consistent with the estimate by
Kataev and Starshenko using a completely different method. We then apply this
improved Borel amplitude to the tau decay width, and obtain the strong coupling
constant $\alpha_s(M_Z) =0.1193 \pm 0.0007_{exp.} \pm 0.0010_{EW+CKM} \pm
0.0009_{meth.} \pm 0.0003_{evol.}$. We then compare this result with those of
other resummation methods.
