  We study the formation of the Milky Way's halo in a $\Lambda$CDM cosmology by
scaling down a high resolution simulation of the formation of a cluster of
galaxies. We determine how much phase-space substructure is left over from the
objects that merge to build up the present galaxy. We study the debris streams
originating from such objects and find that their evolution can be explained
simply in terms of the conservation of phase-space density. Analysing the mass
growth history of our halo we find that its inner regions have been in place
for more than 10 Gyr, but that the growth of the halo as a whole is more
gradual, in agreement with other high resolution simulations of dark-matter
halos. Recent accretion contributes to the inner 10 kpc of the halo only at the
10$^{-4}$ level. Finally we determine the number of dark-matter streams as a
function of distance from the centre of the halo. In the equivalent of the
``Solar vicinity'', we find that the dark-matter is smoothly distributed in
space, and that the velocity ellipsoid is formed by hundreds of thousands of
streams, most of which have velocity dispersions of the order of 1 km/s or
less.
