  Primordial black holes (PBH) may form from large cosmological perturbations,
produced during inflation when the inflaton's velocity is sufficiently slowed
down. This usually requires very flat regions in the inflationary potential. In
this paper we investigate another possibility, namely that the inflaton climbs
up its potential. When it turns back, its velocity crosses zero, which triggers
a short phase of ``uphill inflation'' during which cosmological perturbations
grow at a very fast rate. This naturally occurs in double-well potentials if
the width of the well is close to the Planck scale. We include the effect of
quantum diffusion in this scenario, which plays a crucial role, by means of the
stochastic-$\delta N$ formalism. We find that ultra-light black holes are
produced with very high abundances, which do not depend on the energy scale at
which uphill inflation occurs, and which suffer from substantially less fine
tuning than in alternative PBH-production models. They are such that PBHs later
drive a phase of PBH domination.
