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_{⊙}. 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 unpolarized GW signal, a prediction that is testable with LISA.