Massive galaxies in the early Universe have been shown to be forming stars at surprisingly high rates. Prominent examples are dust-obscured galaxies which are luminous when observed at sub-millimetre wavelengths and which may be forming stars at a rate of 1,000 solar masses (Msolar) per year. These intense bursts of star formation are believed to be driven by mergers between gas-rich galaxies. Probing the properties of individual star-forming regions within these galaxies, however, is beyond the spatial resolution and sensitivity of even the largest telescopes at present. Here we report observations of the sub-millimetre galaxy SMMJ2135-0102 at redshift z = 2.3259, which has been gravitationally magnified by a factor of 32 by a massive foreground galaxy cluster lens. This magnification, when combined with high-resolution sub-millimetre imaging, resolves the star-forming regions at a linear scale of only 100parsecs. We find that the luminosity densities of these star-forming regions are comparable to the dense cores of giant molecular clouds in the local Universe, but they are about a hundred times larger and 107 times more luminous. Although vigorously star-forming, the underlying physics of the star-formation processes at z~2 appears to be similar to that seen in local galaxies, although the energetics are unlike anything found in the present-day Universe.