The chemical reactions responsible for stratospheric ozone depletion are extremely sensitive to temperature(1). Greenhouse gases warm the Earth＇s surface but cool the stratosphere radiatively(2-5) and therefore affect ozone depletion. Here we investigate the interplay between projected future emissions of greenhouse gases and levels of ozone-depleting halogen species using a global climate model that incorporates simplified ozone-depletion chemistry. Temperature and wind changes induced by the increasing greenhouse-gas concentrations alter planetary-wave propagation in our model, reducing the frequency of sudden stratospheric warmings in the Northern Hemisphere(4). This results in a more stable Arctic polar vortex, with significantly colder temperatures in the lower stratosphere and concomitantly increased ozone depletion. Increased concentrations of greenhouse gases might therefore be at least partly responsible for the very large Arctic ozone losses observed in recent winters(6-9). Arctic losses reach a maximum in the decade 2010 to 2019 in our model, roughly a decade after the maximum in stratospheric chlorine abundance. The mean losses are about the same as those over the Antarctic during the early 1990s, with geographically localized losses of up to two-thirds of the Arctic ozone column in the worst years. The severity and the duration of the Antarctic ozone hole are also predicted to increase because of greenhouse-gas-induced stratospheric cooling over the coming decades.