A significant fraction of the energy emitted in the early Universe came from very luminous galaxies that are largely hidden at optical wavelengths (because of interstellar dust grains); this energy now forms part of the cosmic background radiation at wavelengths near 1mm. These submillimetre (submm) galaxies were resolved from the background in 1997 but have been difficult to identify and study due to the poor spatial resolution of submm instruments. This has impeded the determination of their distances (or redshifts, z), a crucial element in understanding their nature and evolution. Here we report spectroscopic redshifts for ten representative submm galaxies that we identified reliably using high resolution radio observations. The median redshift for our sample is 2.4, with a quartile range of z = 1.9-2.8. The submm population therefore coexists with the peak activity of quasars, which are thought to be massive black holes in the process of accreting matter, suggesting a close relationship between the growth of massive black holes and luminous dusty galaxies. The space density of submm galaxies at z>2 is about 1000 times greater than that of similarly luminous galaxies in the present-day Universe, so they represent an important component of star formation at high redshifts.