  Using the method of QCD light-cone sum rules, we calculate the $B \to \pi\pi$
hadronic matrix elements with annihilation topology. We obtain a finite result,
including the related strong phase. Numerically, the annihilation effects in
$B\to \pi\pi$ turn out to be small with respect to the factorizable emission
mechanism. Our predictions, together with the earlier sum rule estimates of
emission and penguin contributions, are used for the phenomenological analysis
of $B\to \pi\pi$ channels. We predict a $\Delta I=1/2$ transition amplitude
which significantly differs from this amplitude extracted from the current
data.
