  We complete the set of QCD light-cone sum rules for $B\to \pi\pi$ transition
form factors, deriving a new sum rule for the timelike-helicity form factor
$F_t$ in terms of dipion distribution amplitudes. This sum rule, in the leading
twist-2 approximation, is directly related to the pion vector form factor.
Employing a relation between $F_t$ and other $B\to \pi\pi$ form factors we
obtain also the longitudinal-helicity form factor $F_0$. In this way, all four
(axial-)vector $B\to \pi\pi$ form factors are predicted from light-cone sum
rules with dipion distribution amplitudes. These results are valid for small
dipion masses with large momentum.
