The discovery of a radioactively powered kilonova associated with the binary neutron star merger GW170817 was the first - and still only - confirmed electromagnetic counterpart to a gravitational-wave event. However, observations of late-time electromagnetic emission are in tension with the expectations from standard neutron-star merger models. Although the large measured ejecta mass is potentially explained by a progenitor system that is asymmetric in terms of the stellar component masses, i.e. with a mass ratio $q$ of 0.7-0.8, the known Galactic population of merging double neutron star (DNS) systems (i.e. those that will coalesce within billions of years or less) has, until now, only consisted of nearly equal-mass ($q > 0.9$) binaries. PSR J1913+1102 is a DNS system in a 5-hour, low-eccentricity ($e = 0.09$) orbit, implying an orbital separation of 1.8 solar radii, with the two neutron stars predicted to coalesce in 470 million years due to gravitational-wave emission. Here we report that the masses of the two neutron stars, as measured by a dedicated pulsar timing campaign, are $1.62 pm 0.03$ and $1.27 pm 0.03$ solar masses for the pulsar and companion neutron star, respectively; with a measured mass ratio $q = 0.78 pm 0.03$, it is the most asymmetric DNS among known merging systems. Based on this detection, our population synthesis analysis implies that such asymmetric binaries represent between 2 and 30% (90% confidence) of the total population of merging DNS 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.
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