  It was recently suggested that the merger of $\sim30\,M_\odot$ primordial
black holes (PBHs) may provide a significant number of events in
gravitational-wave observatories over the next decade, if they make up an
appreciable fraction of the dark matter. Here we show that measurement of the
eccentricities of the inspiralling binary black holes can be used to
distinguish these binaries from those produced by more traditional
astrophysical mechanisms. These PBH binaries are formed on highly eccentric
orbits and can then merge on timescales that in some cases are years or less,
retaining some eccentricity in the last seconds before the merger. This is to
be contrasted with massive-stellar-binary, globular-cluster, or other
astrophysical origins for binary black holes (BBHs) in which the orbits have
very effectively circularized by the time the BBH enters the observable LIGO
window. Here we discuss the features of the gravitational-wave signals that
indicate this eccentricity and forecast the sensitivity of LIGO and the
Einstein Telescope to such effects. We show that if PBHs make up the dark
matter, then roughly one event should have a detectable eccentricity given
LIGO's expected sensitivity and observing time of six years. The Einstein
Telescope should see $O(10)$ such events after ten years.
