We present a complete sample of megaparsec-sized double radio sources compiled from the Sydney University Molonglo Sky Survey. Almost complete redshift information has been obtained for the sample. The sample has the following defining criteria: Galactic latitude > 12fdg5, declination δ < -50°, and angular size >5'. All the sources have a projected linear size larger than 0.7 Mpc (assuming H0 = 71 km s-1 Mpc-1). The sample is chosen from a region of the sky covering 2100 deg2. In this paper we present 843 MHz radio images of the extended radio morphologies made using the Molonglo Observatory Synthesis Telescope, higher resolution radio observations of any compact radio structures using the Australia Telescope Compact Array, and low-resolution optical spectra of the host galaxies from the 2.3 m Australian National University telescope at Siding Spring Observatory. The sample presented here is the first in the southern hemisphere and significantly enhances the database of known giant radio sources. The giant radio sources with linear size exceeding 0.7 Mpc have an abundance of (215 Mpc)-3 at the sensitivity of the survey. In the low-redshift universe, the survey may be suggesting the possibility that giant radio sources with relict lobes are more numerous than giant sources in which beams from the center currently energize the lobes.