The production of 22 Na in ONe novae can be influenced by the 22 Mg(p,γ ) 23 Al reaction. To investigate this reaction rate at stellar energies, we have determined the asymptotic normalization coefficient (ANC) for 22 Mg + p → 23 Al through measurements of the ANCs in the mirror nuclear system 22 Ne + n → 23 Ne. The peripheral neutron-transfer reactions 13 C( 12 C, 13 C) 12 C and 13 C( 22 Ne, 23 Ne) 12 C were studied. The identical entrance and exit channels of the first reaction make it possible to extract independently the ground-state ANC in 13 C. Our experiment gives C 2 p 1/2 ( 13 C) = 2.24 ± 0.11 fm −1 , which agrees with the value obtained from several previous measurements. The weighted average for all the obtained C 2 p 1/2 is 2.31 ± 0.08 fm −1 . This value is adopted to be used in obtaining the ANCs in 23 Ne. The differential cross sections for the reaction 13 C( 22 Ne, 23 Ne) 12 C leading to the J π = 5/2 + and 1/2 + states in 23 Ne have been measured at 12 MeV/u. Optical model parameters for use in the DWBA calculations were obtained from measurements of the elastic scatterings 22 Ne + 13 C and 22 Ne + 12 C. The extracted ANC for the ground state in 23 Ne, C 2 d 5/2 = 0.86 ± 0.08 ± 0.12 fm −1 , is converted to its corresponding value in 23 Al using mirror symmetry to give C 2 d 5/2 ( 23 Al) = (4.63 ± 0.77) × 10 3 fm −1 . The astrophysical S factor S(0) for the 22 Mg(p,γ ) reaction was determined to be 0.96 ± 0.11 keV b. The consequences for nuclear astrophysics are discussed.