PPI 15 is the first object to have been confirmed as a brown dwarf by the l
ithium test (in 1995), though its inferred mass was very close to the subst
ellar limit. It is a member of the Pleiades open cluster. Its position in a
cluster color-magnitude diagram suggested that it might be binary, and pre
liminary indications that it is a double-lined spectroscopic binary were re
ported by us in 1997. Here we report on the results of a consecutive week o
f Keck HIRES observations of this system, which yield its orbit. It has a p
eriod of about 5.8 days, and an eccentricity of 0.4 +/- 0.05. The rotation
of the stars is slow for this class of objects. Because the system luminosi
ty is divided between two objects with a mass ratio of 0.85, each of them i
s rendered an incontrovertible brown dwarf, with masses between 60 and 70 M
-J. We show that component B is a little redder than A by studying their wa
velength-dependent line ratios and that this variation is compatible with t
he mass ratio. We confirm that the system has lithium but cannot support th
e original conclusion that it is depleted (which would be surprising, given
the new masses). This is a system of very close objects, which, if they ha
d combined, would have produced a low-mass star. We discuss the implication
s of this discovery for the theories of binary formation and formation of v
ery low mass objects.