  We determine the binding energy, the total gravitational wave energy flux,
and the gravitational wave modes for a binary of rapidly spinning black holes,
working in linearized gravity and at leading orders in the orbital velocity,
but to all orders in the black holes' spins. Though the spins are treated
nonperturbatively, surprisingly, the binding energy and the flux are given by
simple analytical expressions which are finite (respectively third- and
fifth-order) polynomials in the spins. Our final results are restricted to the
important case of quasi-circular orbits with the black holes' spins aligned
with the orbital angular momentum.
