We reanalyze the decay $B_d\to\pi^+\pi^-$ without assuming dominance
of QCD penguins with internal top-quark exchanges. In that case the
weak phase of the $\bar b \to\bar d$ QCD penguin amplitude is not
related in a simple way to the CKM angle $\beta$. Nevertheless it is
still possible to extract the CKM angle $\alpha$ in a transparent and
almost model-independent way from the time-dependent CP-violating
asymmetry of the transition $B_d\to\pi^+\pi^-$. Aside from that
CP asymmetry this approach needs as an input only amplitudes of decays
with branching ratios of order $10^{-5}$ and will thus be well within
reach at future $B$-factories.
