Antihydrogen atoms may rarely be produced by the collision of antiprotons with ions. At relativistic velocities, the antiproton may pass around the Coulomb field of the nucleus and the electron-positron pairs can be produced electromagnetically. After this pair production, not so often, positron can be captured by the antiproton and as a result, antihydrogen atoms may be produced. In this work, we have calculated the antihydrogen production cross section in the framework of perturbation theory, by applying Monte-Carlo integration techniques. In order to compute the lowest-order Feynman diagrams amplitudes, we used Darwin wave functions for the bound states of the positrons and Sommerfeld-Maue wave functions for the continuum states of the electrons. (C) 2012 Elsevier B.V. All rights reserved.