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 .