In the era of precision cosmology, it is essential to determine the Hubble constant to an accuracy of three per cent or better(1,2). At present, its uncertainty is dominated by the uncertainty in the distance to the Large Magellanic Cloud (LMC), which, being our second-closest galaxy, serves as the best anchor point for the cosmic distance scale(2,3). Observations of eclipsing binaries offer a unique opportunity to measure stellar parameters and distances precisely and accurately(4,5). The eclipsing-binary method was previously applied to the LMC6,7, but the accuracy of the distance results was lessened by the need to model the bright, early-type systems used in those studies. Here we report determinations of the distances to eight long-period, late-type eclipsing systems in the LMC, composed of cool, giant stars. For these systems, we can accurately measure both the linear and the angular sizes of their components and avoid the most important problems related to the hot, early-type systems. The LMC distance that we derive from these systems (49.97 +/- 0.19 (statistical) +/- 1:11 (systematic) kiloparsecs) is accurate to 2.2 per cent and provides a firm base for a 3-per-cent determination of the Hubble constant, with prospects for improvement to 2 per cent in the future.