The 95 Zr(n,γ) 96 Zr reaction cross section is crucial in the modelling of s-process nucleosynthesis in asymptotic giant branch stars because it controls the operation of the branching point at the unstable 95 Zr and the subsequent production of 96 Zr. We have carried out the measurement of the 94 Zr( 18 O, 16 O) and 90 Zr( 18 O, 16 O) reactions and obtained the γ-decay probability ratio of 96 Zr* and 92 Zr* to determine the 95 Zr(n,γ) 96 Zr reaction cross sections with the surrogate ratio method. Our deduced maxwellian-averaged cross section of 66±16 mb at 30 keV is close to the value recommended by Bao et al. (2000), but 30% and more than a factor of two larger than the values proposed by Toukan & Käppeler (1990) and Lugaro et al. (2014), respectively, and routinely used in s-process models. We tested the new rate in stellar models with masses between 2 and 6 M ⊙ and metallicities 0.014 and 0.03. The largest changes -up 80% variations in 96 Zr -are seen in models of mass 3-4 M ⊙ , where the 22 Ne neutron source is mildly activated. The new rate can still provide a match to data from meteoritic stardust silicon carbide grains, provided the maximum mass of the parent stars is below 4 M ⊙ , for a metallicity of 0.03.