  A significant fraction of cosmological dark matter can be formed by very
dense macroscopic objects, for example primordial black holes. Gravitational
waves offer a promising way to probe these kinds of dark-matter candidates, in
a parameter space region that is relatively untested by electromagnetic
observations. In this work we consider an ensemble of macroscopic dark matter
with masses in the range $10^{-13}$ - $1\,M_{\odot}$ orbiting a super-massive
black hole. While the strain produced by an individual dark-matter particle
will be very small, gravitational waves emitted by a large number of such
objects will add incoherently and produce a stochastic gravitational-wave
background. We show that LISA can be a formidable machine for detecting the
stochastic background of such objects orbiting the black hole in the centre of
the Milky Way, Sgr\.${\rm A}^{\!*}$, if a dark-matter spike of the type
originally predicted by Gondolo and Silk forms near the central black hole.
