The production of antihydrogen in flight in antiproton-nucleus collisions is calculated theoretically in the Plane Wave Born Approximation (which is equivalent to the straight line semiclassical approximation). Antihydrogen has been produced in this way at LEAR/CERN and is presently studied at FERMILAB at various antiproton energies. Dirac wave functions for the leptons are used, taking first order (Z.alpha) corrections into account. Analytical results are obtained for differential cross-sections. Total cross sections are obtained by numerical integration. The dependence on the transverse momentum transfer is studied and the accuracy of the equivalent photon approximation and a recent variant by Munger, Brodsky,and Schmidt is discussed as a function of beam energy.