Satellite data are presented showing the timing of sudden warmings in the lower stratosphere during the winters 1979--1992. A three-dimensional dynamical-radiative-photochemical model is used to establish how Arctic ozone depletion will respond to a doubling of CO2 according to the timing of the warmings. In a series of idealized experiments the timing of the warmings is varied by specifying different geopotential wave amplitudes at the 316-mbar model lower boundary. Results from a ''transient climate change experiment'' show that the chosen wave amplitudes are appropriate for both the current and the doubled CO2 atmosphere. For doubled CO2 the experiments show that any significant risk of an Arctic ozone hole will be confined to those years with only a late stratospheric warming. In all other years the results suggest that springtime total ozone over the Arctic is more likely to increase by a small amount due to a combination of slower homogeneous chemistry and changes in transport. The predictions obtained from the idealized studies are then tested by prescribing at the model lower boundary the observed geopotential wave amplitudes from two specific years with late winter warmings. Doubling CO2 amounts produced no significant increase in ozone depletion with the 1989 wave amplitudes, but with 1990 wave amplitudes, an Arctic ozone hole occurred with minimum column of 187 Dobson Units. This contrasting response is attributed to the large midwinter pulse in the 1989 wave amplitudes compared to the less dramatic and shorter timescale fluctuations in the 1990 wave amplitudes. It is concluded that under doubled CO2 conditions an Arctic ozone hole is likely to occur in years with late stratospheric warmings following winters in which there were no significant pulses in the upper tropospheric planetary wave amplitudes.