We report here the detection of a weak magnetic field of 50 - 100 G on the O9.7 supergiant zeta Ori A, using spectropolarimetric observations obtained with NARVAL at the 2m Telescope Bernard Lyot atop Pic du Midi (France). zeta Ori A is the third O star known to host a magnetic field (along with theta^1 Ori C and HD 191612), and the first detection on a normal rapidly-rotating O star. The magnetic field of zeta Ori A is the weakest magnetic field ever detected on a massive star. The measured field is lower than the thermal equipartition limit (about 100 G). By fitting NLTE model atmospheres to our spectra, we determined that zeta Ori A is a 40 Msun star with a radius of 25 Rsun and an age of about 5 - 6 Myr, showing no surface nitrogen enhancement and losing mass at a rate of about 2x10^(-6) Msol/yr. The magnetic topology of zeta Ori A is apparently more complex than a dipole and involves two main magnetic polarities located on both sides of the same hemisphere; our data also suggest that zeta Ori A rotates in about 7.0 d and is about 40 degrees away from pole-on to an Earth-based observer. Despite its weakness, the detected magnetic field significantly affects the wind structure; the corresponding Alfven radius is however very close to the surface, thus generating a different rotational modulation in wind lines than that reported on the two other known magnetic O stars. The rapid rotation of zeta Ori A with respect to theta^1 Ori C appears as a surprise, both stars having similar unsigned magnetic fluxes (once rescaled to the same radius); it may suggest that the sub-equipartition field detected on zeta Ori A is not a fossil remnant (as opposed to that of theta^1 Ori C and HD 191612), but the result of an exotic dynamo action produced through MHD instabilities.
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