The Hubble Deep Field provides one of the deepest multiwavelength views of the distant Universe and has led to the detection of thousands of galaxies seen throughout cosmic time. An early map of the Hubble Deep Field at a wavelength of 850 micrometres, which is sensitive to dust emission powered by star formation, revealed the brightest source in the field, dubbed HDF 850.1 (ref. 2). For more than a decade, and despite significant efforts, no counterpart was found at shorter wavelengths, and it was not possible to determine its redshift, size or mass. Here we report a redshift of z = 5.183 for HDF 850.1, from a millimetre-wave molecular line scan. This places HDF 850.1 in a galaxy overdensity at z ≈ 5.2, corresponding to a cosmic age of only 1.1 billion years after the Big Bang. This redshift is significantly higher than earlier estimates and higher than those of most of the hundreds of submillimetre-bright galaxies identified so far. The source has a star-formation rate of 850 solar masses per year and is spatially resolved on scales of 5 kiloparsecs, with an implied dynamical mass of about 1.3 × 10(11) solar masses, a significant fraction of which is present in the form of molecular gas. Despite our accurate determination of redshift and position, a counterpart emitting starlight remains elusive.