Position angles and magnitudes of linear polarization in the H band have been measured for 66 stars embedded in or behind the Chamaeleon I dark cloud. These data are used to demonstrate that the magnetic field permeating the cloud has an ordered structure over the full extent of the dark cloud (~4pc), with the magnetic field direction changing smoothly from a position angle of ~100 degrees in the vicinity of HD 97300 in the north to ~140 degrees in the vicinity of HD 97048 in the south and west. The elongation of the dark cloud is consistent with collapse preferentially along magnetic field lines. The polarizing efficiency of the cloud decreases with increasing line-of-sight extinction. Cloud members show a larger range in degree of polarization and position angle, which is interpreted as due to anisotropic scattering local to these stars. The IRAS 11054-7706C bipolar CO outflow is unusual in that it is oriented at >60 degrees to the cloud magnetic field direction. The Chamaeleon Infrared Nebula is polarized at ~30 per cent in the H band and also is oriented at ~50 degrees to the local magnetic field. We suggest that gravitational forces may have overwhelmed magnetic support locally to form these relatively massive stars.