  Despite their importance in the star formation process, measurements of
magnetic field strength in proto-planetary discs remain rare. While linear
polarisation of dust and molecular lines can give insight into the magnetic
field structure, only observations of the circular polarisation produced by
Zeeman splitting provide a direct measurement of magnetic field strenghts. One
of the most promising probes of magnetic field strengths is the paramagnetic
radical CN. Here we present the first Atacama Large Millimeter/submillimeter
Array (ALMA) observations of the Zeeman splitting of CN in the disc of TW~Hya.
The observations indicate an excellent polarisation performance of ALMA, but
fail to detect significant polarisation. An analysis of eight individual CN
hyperfine components as well as a stacking analysis of the strongest
(non-blended) hyperfine components yields the most stringent limits obtained so
far on the magnetic field strength in a proto-planetary disc. We find that the
vertical component of the magnetic field $|B_z|<0.8$~mG ($1\sigma$ limit). We
also provide a $1\sigma$ toroidal field strength limit of $<30$~mG. These
limits rule out some of the earlier accretion disc models, but remain
consistent with the most recent detailed models with efficient advection. We
detect marginal linear polarisation from the dust continuum, but the almost
purely toroidal geometry of the polarisation vectors implies that his is due to
radiatively aligned grains.
