Redshifts are presented for a magnitude-limited sample (bJ ≤ 16.5) of 107 galaxies detected from scans with the COSMOS machine in the central 29 square degrees of the ESO/SERC southern sky survey field 349 (centred at 00h 00m, -35° 00', 1950 coordinates). Spectra at a dispersion of ~210 Å mm-1 were obtained with the 1.9-m telescope of the South African Astronomical Observatory (SAAO), using the Image-tube Spectrograph and Reticon Photon Counting System (RPCS). Both cross-correlation techniques against filtered templates and direct measures of emission lines (if present) were used to obtain radial velocities to an accuracy of ~75 km s-1. The observed sample represents a completeness of ~80 per cent for all the galaxies detected by COSMOS down to bJ = 16.5 and provides a significant data base for analysis of the brighter end of the galaxy distribution in this field. The remaining unobserved galaxies were generally fainter than bJ = 16.0 although three were of low surface brightness. This survey, going ~2.0 mag fainter than the CfA survey, is compatible with the Durham/AAT redshift survey (Shanks et al.) and should provide further insight into the nature of galaxy clustering.