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  1. ABSTRACT Two binary neutron star mergers, GW170817 and GW190425, have been detected by Advanced LIGO and Virgo. These signals were detected by matched-filter searches that assume that the star’s orbit has circularized by the time their gravitational-wave emission is observable. This suggests that their eccentricity is low, but full parameter estimation of their eccentricity has not yet been performed. We use gravitational-wave observations to measure the eccentricity of GW170817 and GW190425. We find that the eccentricity at a gravitational-wave frequency of 10 Hz is e ≤ 0.024 and e ≤ 0.048 for GW170817 and GW190425, respectively (90 per cent confidence). This is consistent with the binaries being formed in the field, as such systems are expected to have circularized to e ≤ 10−4 by the time they reach the LIGO–Virgo band. Our constraint is a factor of 2 smaller that an estimate based on GW170817 being detected by searches that neglect eccentricity. However, we caution that we find significant prior dependence in our limits, suggesting that there is limited information in the signals. We note that other techniques used to constrain binary neutron star eccentricity without full parameter estimation may miss degeneracies in the waveform, and that for future signals, it will bemore »important to perform full parameter estimation with accurate waveform templates.« less