  There has recently been renewed interest in the possibility that the dark
matter in the universe consists of primordial black holes (PBHs). Current
observational constraints leave only a few PBH mass ranges for this
possibility. One of them is around $10^{-12} M_\odot$. If PBHs with this mass
are formed due to an enhanced scalar-perturbation amplitude, their formation is
inevitably accompanied by the generation of gravitational waves (GWs) with
frequency peaked in the mHz range, precisely around the maximum sensitivity of
the LISA mission. We show that, if these primordial black holes are the dark
matter, LISA will be able to detect the associated GW power spectrum. Although
the GW source signal is intrinsically non-Gaussian, the signal measured by LISA
is a sum of the signal from a large number of independent sources suppressing
the non-Gaussianity at detection to an unobservable level. We also discuss the
effect of the GW propagation in the perturbed universe. PBH dark matter
generically leads to a detectable, purely isotropic, Gaussian and unpolarised
GW signal, a prediction that is testable with LISA.
