  We study the production of primordial black hole (PBH) dark matter in the
case when the Standard Model Higgs coupled non-minimally to gravity is the
inflaton. PBHs can be produced if the Higgs potential has a near-critical point
due to quantum corrections. In this case the slow-roll approximation may be
broken, so we calculate the power spectrum numerically. We consider both the
metric and the Palatini formulation of general relativity. Combining
observational constraints on PBHs and on the CMB spectrum we find that PBHs can
constitute all of the dark matter only if they evaporate early and leave behind
Planck mass relics. This requires the potential to have a shallow local
minimum, not just a critical point. The initial PBH mass is then below $10^6$
g, and predictions for the CMB observables are the same as in tree-level Higgs
inflation, $n_s=0.96$ and $r=5\times10^{-3}$ (metric) or $r=4\times 10^{-8}
\dots 2 \times 10^{-7}$ (Palatini).
