  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 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 $e \leq 0.024$ and $e \leq 0.048$ for
GW170817 and GW190425, respectively (90% confidence). This is consistent with
the binaries being formed in the field, as such systems are expected to have
circularized to $e \leq 10^{-4}$ by the time they reach the LIGO-Virgo band.
Our constraint is a factor of two 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 be
important to perform full parameter estimation with accurate waveform
templates.
