The electron impact excitation cross section for the resonance transition of Mg II is calculated in a four-state close-coupling approximation. The interaction Hamiltonian contains a model potential that accounts for core polarisation effects. A detailed analysis of the resonance series converging to the 4s 2S and 3d 2D thresholds is made using quantum defect theory techniques. The total cross section is compared with experiment and other calculations, and rate coefficients are presented for electron temperatures of 5000-20000K. Cross sections are also calculated for the transitions 3s-3d, 3p-3d, 3s-4s, 3p-4s, 4s-3d and 3p 2P12/0-3p 2P32/0.