The fate of an evaporating black hole when it reaches the Planck mass is a matter for conjecture. Here, we propose that the cosmological dark matter consists of the Planck-mass remnants of evaporating primordial black holes. Such remnants would be expected to have close to the critical density if the black holes evaporating at the present epoch have the maximum density consistent with cosmic-ray constraints. The remnants are also candidates for the missing mass in the galactic halo. Primordial black holes of the required density may form naturally at the end of an inflationary epoch. Planck-mass relics would behave dynamically just like 'cold dark matter' and would therefore share the attractions of other 'cold' candidates. In addition, because the baryonic matter in black holes cannot participate in nucleosynthesis the limits on the baryonic content of the Universe set by primordial nucleosynthesis are circumvented.