The $^{22}$Ne(α,γ)$^{26}$Mg and $^{22}$Ne(α,n)$^{25}$Mg reactions play an important role in astrophysics because they have significant influence on the neutron flux during the weak branch of the s-process. We constrain the astrophysical rates for these reactions by measuring partial α-widths of resonances in $^{26}$Mg located in the Gamow window for the ${}^{22}\mathrm{Ne}+\alpha$ capture. These resonances were populated using $^{22}$Ne($^{6}$Li,d)$^{26}$Mg and $^{22}$Ne($^{7}$Li,t)$^{26}$Mg reactions at energies near the Coulomb barrier. At these low energies α-transfer reactions favor population of low spin states and the extracted partial α-widths for the observed resonances exhibit only minor dependence on the model parameters. The astrophysical rates for both the $^{22}$Ne(α,γ)$^{26}$Mg and the $^{22}$Ne(α,n)$^{25}$Mg reactions are shown to be significantly different than the previously suggested values.