We present new radial velocities for 306 bright (R < 16) galaxies in a 77 deg2 region of the Shapley supercluster, measured with the FLAIR-II spectrograph on the UK Schmidt Telescope. The galaxies we measured were uniformly distributed over the survey area, in contrast to previous samples which were concentrated in several rich Abell clusters. Most of the galaxies (230) were members of the Shapley supercluster: they trace out two previously unknown sheets of galaxies linking the Abell clusters of the supercluster. In a 44 deg2 area of the supercluster excluding the Abell clusters, these sheets alone represent an overdensity of a factor of 2·0 ± 0·2 compared to a uniform galaxy distribution. The supercluster is not flattened in the Declination direction as has been suggested in previous papers. Within our survey area the new galaxies contribute an additional 50% to the known contents of the Shapley supercluster, with a corresponding increase in its contribution to the motion of the Local Group.