We have used the Australia Telescope Compact Array (ATCA) to search for faint radio sources in a ~3deg^2^ region of sky covered by the 2dF Galaxy Redshift Survey (2dFGRS, Cat. <VII/226>). Over the region surveyed, the 1{sigma} noise level at 1.4GHz ranges from 20Jy to 1mJy. The survey region includes 365 2dFGRS galaxies, of which 316 have good-quality spectra (176 early-type galaxies or active galactic nuclei, and 140 star-forming galaxies). The fraction of 2dFGRS galaxies detected as radio sources in our survey rises from ~4% at a 3{sigma} detection limit of 0.3mJy to 12% at 75{mu}Jy, with roughly equal numbers of star-forming galaxies and active galactic nuclei (AGNs) being detected.