To understand the evolution of galaxies, we need to know as accurately as possible how many galaxies were present in the Universe at different epochs. Galaxies in the young Universe have hitherto mainly been identified using their expected optical colours, but this leaves open the possibility that a significant population remains undetected because their colours are the result of a complex mix of stars, gas, dust or active galactic nuclei. Here we report the results of a flux-limited I-band survey of galaxies at look-back times of 9 to 12 billion years. We find 970 galaxies with spectroscopic redshifts between 1.4 and 5. This population is 1.6 to 6.2 times larger than previous estimates, with the difference increasing towards brighter magnitudes. Strong ultraviolet continua (in the rest frame of the galaxies) indicate vigorous star formation rates of more than 10-100 solar masses per year. As a consequence, the cosmic star formation rate representing the volume-averaged production of stars is higher than previously measured at redshifts of 3 to 4.