  We determine the ability of Cosmic Explorer, a proposed third-generation
gravitational-wave observatory, to detect eccentric binary neutron stars and to
measure their eccentricity. We find that for a matched-filter search, template
banks constructed using binaries in quasi-circular orbits are effectual for
eccentric neutron star binaries with $e_{7} \leq 0.004$ ($e_{7} \leq 0.003$)
for CE1 (CE2), where $e_7$ is the binary's eccentricity at a gravitational-wave
frequency of 7~Hz. We show that stochastic template placement can be used to
construct a matched-filter search for binaries with larger eccentricities and
construct an effectual template bank for binaries with $e_{7} \leq 0.05$. We
show that the computational cost of both the search for binaries in
quasi-circular orbits and eccentric orbits is not significantly larger for
Cosmic Explorer than for Advanced LIGO and is accessible with present-day
computational resources. We investigate Cosmic Explorer's ability to
distinguish between circular and eccentric binaries. We estimate that for a
binary with a signal-to-noise ratio of 8 (800), Cosmic Explorer can distinguish
between a circular binary and a binary with eccentricity $e_7 \gtrsim 10^{-2}$
($10^{-3}$) at 90% confidence.
