Abstract: Waste from a nuclear power plant, containing organic ion-exchange resin containing radioactive metals, is decomposed to form volatile substances by means of oxidation in an acid, aqueous liquid (8). Radioactive metal ions thus dissolved in the liquid, which are precipitable with hydroxide, are then caused to precipitate by the addition of a hydroxide (10) at a pH value of at least 9, and the metal compounds thus precipitated are separated from at least most of the liquid. Radioactive caesium and strontium ions in this liquid are removed therefrom. The liquid (24) separated from precipitated metal compounds and freed from caesium and strontium ions is released to a recipient, with the remaining substances still dissolved in it, and the precipitated metal compounds (11k, 11s) are solidified in a deposition container (22). The ion-exchange resin preferably consists at least partially of sulphonated ion-exchange resin and the hydroxide preferably of sodium hydroxide or some other alkali metal hydroxide.