Neutron destruction reactions of the cosmic γ-ray emitter Al26 are of importance to determine the amount of Al26 ejected into our galaxy by supernova explosions and for Al26 production in asymptotic giant branch stars. We performed a new measurement of the Al26(n,α) reaction up to 160-keV neutron energy at the neutron time-of-flight facilities n_TOF at CERN and GELINA at EC-JRC. We provide strengths for ten resonances, six of them for the first time. We use our data to calculate astrophysical reactivities for stellar temperatures up to 0.7 GK. Our results resolve a discrepancy between the two previous direct measurements of this reaction, and indicate higher stellar destruction rates than the most recently recommended reactivity.