  A simple Born-form approximation at the one-loop level for e^+e^- to W^+ W^-
at high energies is given in analytic form. The different contributions to the
three invariant one-loop amplitudes, S_I^{(-)}(s,t) and S_Q^{(\pm)}(s,t),
determining the Born-form helicity amplitudes are thoroughly investigated
analytically and numerically. At energies above 500 GeV, the accuracy of the
simple Born-form approximation for the differential production cross section is
better than 1% for almost all W^+W^- production angles, independently of
whether the W^+W^- polarization is summed over, or whether a longitudinal or a
transverse polarization is selected for both the W^+ and the W^-.
