  We present the first far infrared (FIR) dust emission polarization map
covering the full extent Milky Way's Central molecular zone (CMZ). The data,
obtained with the PILOT balloon-borne experiment, covers the Galactic Center
region $-2\,^\circ<l<2\,^\circ$, $-4\,^\circ<b<3\,^\circ$ at a wavelength of
240 $\mu$m and an angular resolution $2.2\,'$. From our measured dust
polarization angles, we infer a magnetic field orientation projected onto the
plane of the sky that is remarkably ordered over the full extent of the CMZ,
with an average tilt angle of $\simeq 22\,^\circ$ clockwise with respect to the
Galactic plane. Our results confirm previous claims that the field traced by
dust polarized emission is oriented nearly orthogonal to the field traced by
GHz radio synchrotron emission in the Galactic Center region. The observed
field structure is globally compatible with the latest Planck polarization data
at 353 GHz and 217 GHz. Upon subtraction of the extended emission in our data,
the mean field orientation that we obtain shows good agreement with the mean
field orientation measured at higher angular resolution by the JCMT within the
20 km/s and 50 km/s molecular clouds. We find no evidence that the magnetic
field orientation is related to the 100 pc twisted ring structure within the
CMZ. We propose that the low polarization fraction in the Galactic Center
region and the highly ordered projected field orientation can be reconciled if
the field is strong, with a 3D geometry that is is mostly oriented $\simeq
15\,^\circ$ with respect to the line-of-sight towards the Galactic center.
Assuming equipartition between the magnetic pressure and ram pressure, we
obtain magnetic field strengths estimates as high as a few mG for several CMZ
molecular clouds.
