  We have compiled a sample of 16 orbits for very low-mass stellar (<0.1 Msun)
and brown dwarf binaries, enabling the first comprehensive study of the
eccentricity distribution for such objects. We find that very low-mass binaries
span a broad range of eccentricities (0.03<e<0.83), with a median eccentricity
of 0.34. We examine potential observational biases in this sample, and for
visual binaries we show through Monte Carlo simulations that appropriate
selection criteria result in all eccentricities being equally represented (<5%
difference between input and output e distributions). The orbits of this sample
of very low-mass binaries show some significant differences from their
solar-type counterparts. They lack a correlation between orbital period and
eccentricity and display a much higher fraction of near-circular orbits (e<0.1)
than solar-type stars, which together suggest a different formation mechanism
or dynamical history for these two populations. Very low-mass binaries also do
not follow the e^2 distribution of Ambartsumian (1937), which would be expected
if their orbits were distributed in phase space according to a function of
energy alone (e.g., the Boltzmann distribution). We find that numerical
simulations of very low-mass star formation do not completely reproduce the
observed properties of our binary sample. The cluster formation model of Bate
(2009) agrees very well with the overall e distribution, but lack any high-e
(>0.6) binaries at orbital periods comparable to our sample. In contrast, the
circumstellar disk fragmentation model of Stamatellos & Whitworth (2009)
predicts only high-e binaries and thus is highly inconsistent with our sample.
These discrepancies could be explained if multiple formation processes have
produced the field population.
