  Aims. The main-sequence B-type star $\zeta$ Cassiopeiae is known as a N-rich
star with a magnetic field discovered with the Musicos spectropolarimeter. We
model the magnetic field of the star by means of 82 new spectropolarimetric
observations of higher precision to investigate the field strength, topology,
and effect.
  Methods. We gathered data with the Narval spectropolarimeter installed at
T\'elescope Bernard Lyot (TBL, Pic du Midi, France) and applied the
least-squares deconvolution technique to measure the circular polarisation of
the light emitted from $\zeta$ Cas. We used a dipole oblique rotator model to
determine the field configuration by fitting the longitudinal field
measurements and by synthesizing the measured Stokes V profiles. We also made
use of the Zeeman-Doppler Imaging technique to map the stellar surface and to
deduce the difference in rotation rate between the pole and equator.
  Results. $\zeta$ Cas exhibits a polar field strength $B_{\rm pol}$ of 100-150
G, which is the weakest polar field observed so far in a massive main-sequence
star. Surface differential rotation is ruled out by our observations and the
field of $\zeta$ Cas is strong enough to enforce rigid internal rotation in the
radiative zone according to theory. Thus, the star rotates as a solid body in
the envelope.
  Conclusions. We therefore exclude rotationally-induced mixing as the cause of
the surface N-enrichment. We discuss that the transport of chemicals from the
core to the surface by internal gravity waves is the most plausible explanation
for the nitrogen overabundance at the surface of $\zeta$ Cas.
