We have carried out observations at 18.5 GHz with the Australia Telescope Compact Array (ATCA) of 250 out of the 258 Southern extragalactic sources in the complete 5 GHz 1 Jy sample by K\"uhr et al. (1981). In this paper we focus on the polarization properties of this sample, while other properties will be addressed in a future paper. In our analysis we subdivide the sample into flat and steep spectrum sources following Stickel et al. (1994) classification, where spectral indices were measured between 2.7 and 5 GHz. The polarized flux has been measured with a S/N > 5 for 170 sources (114 flat-spectrum and 56 steep-spectrum) and upper limits have been set for additional 27 sources (12 flat-spectrum and 15 steep-spectrum). The median polarization degree at 18.5 GHz for the flat-spectrum sub-sample is Π_{18.5}≃ 2.7%, about a factor of 2 higher than at 1.4 GHz (Π_{1.4}≃ 1.4%, based on NVSS data). For flat-spectrum sources we find a weak correlation between Π_{18.5} and the high frequency (5--18.5 GHz) spectral index. No evidences of significant correlations of the polarization degree with other source properties are found. The median value of Π_{18.5} for the steep spectrum sources is ≃ 4.8%, but our sample might be biased against extended sources. We find indications of a correlation between Π_{18.5} and both the low frequency (1.4--5 GHz) and the high frequency (5--18.5 GHz) spectral indices. An important application of this work is the possibility to estimate the contamination of CMB polarization maps by extragalactic radio sources. Our results indicate that such contamination is within the range of estimates given by Mesa et al. (2002). Comment: 11 pages, 8 Postscripts figures, 6 tables, accepted for publication on A&A, minor typos corrected