The discovery of a radioactively powered kilonova associated with the binary neutron-star merger GW170817 remains the only confirmed electromagnetic counterpart to a gravitational-wave event(1,2). Observations of the late-time electromagnetic emission, however, do not agree with the expectations from standard neutron-star merger models. Although the large measured ejecta mass(3,4 )could be explained by a progenitor system that is asymmetric in terms of the stellar component masses (that is, with a mass ratio q of 0.7 to 0.8)(5), the known Galactic population of merging double neutron-star systems (that is, those that will coalesce within billions of years or less) has until now consisted only of nearly equal-mass (q> 0.9) binaries(6). The pulsar PSR J1913+1102 is a double system in a five-hour, low-eccentricity (0.09) orbit, with an orbital separation of 1.8 solar radii(7), and the two neutron stars are predicted to coalesce in 470(-11)(+12) million years owing to gravitational-wave emission. Here we report that the masses of the pulsar and the companion neutron star, as measured by a dedicated pulsar timing campaign, are 1.62 +/- 0.03 and 1.27 +/- 0.03 solar masses, respectively. With a measured mass ratio of q= 0.78 +/- 0.03, this is the most asymmetric merging system reported so far. On the basis of this detection, our population synthesis analysis implies that such asymmetric binaries represent between 2 and 30 per cent (90 per cent confidence) of the total population of merging binaries. The coalescence of a member of this population offers a possible explanation for the anomalous properties of GW170817, including the observed kilonova emission from that event.