The recent periastron passage of the pulsar PSR B1259-63 around its companion star SS 2883 gave a unique opportunity to study the characteristics of the disc of a Be star. We report here on extensive multifrequency radio observations of the pulsar from 1993 August to 1994 October, which show large changes in its flux density, linear polarization, dispersion measure (DM) and rotation measure (RM), often on short time-scales. For five weeks centred on the periastron epoch of 1994 January 9 (MJD 49361.2), no pulsed flux was detected. This is most likely due to a combination of free-free absorption and severe pulse scattering in the Be-star disc, even at frequencies as high as 8.4 GHz. In the week before the eclipse, the DM increased by ~10 cm^-3 pc and the amount of pulse scattering rapidly increased. Following the re-appearance of the pulsed emission in 1994 February, the DM had returned to normal, but the RM had changed dramatically to -6600 rad m^-2 from its nominal value of +21 rad m^-2. By 1994 April it had gradually returned to its normal value. Using a simple model for a Be-star disc, we conclude that the density of the disc is more than 10^12 cm^-3 at the surface of the star, with a steep radial dependence, and that the magnetic field at the surface of the star is about 1G.