  In a system of particles, quasi-periodic almost-collision orbits are
collisionless orbits along which two bodies become arbitrarily close to each
other -- the lower limit of their distance is zero but the upper limit is
strictly positive -- and which are quasi-periodic in a regularized system up to
a change of time. The existence of such orbits was shown in the restricted
planar circular three-body problem by A. Chenciner and J. Llibre, and later, in
the planar three-body problem by J. F\'ejoz. In the spatial three-body problem,
the existence of a set of positive measure of such orbits was predicted by C.
Marchal. In this article, we present a proof of this fact.
