We describe here changes in stratospheric dynamics and chemistry in a first century‐long sulfate aerosol geoengineering simulation in which the mean surface temperature and the interhemispheric and equator‐to‐pole surface temperature gradients were kept near their 2020 levels despite the RCP8.5 emission scenario. Simulations were carried out with the Community Earth System Model, version 1 with the Whole Atmosphere Community Climate Model as its atmospheric component [CESM1(WACCM)] coupled to a feedback algorithm controlling the magnitude of sulfur dioxide (SO_2) injections at four injection latitudes. We find that, throughout the entire geoengineering simulation, the lower stratospheric temperatures increase by ∼0.19 K per Tg SO_2 injection per year or ∼10 K with ∼40 Tg SO_2/year total SO_2 injection. These temperature changes are associated with a strengthening of the polar jets in the stratosphere and weakening of the mean zonal wind in the lower stratosphere subtropics and throughout the troposphere, associated with weaker storm track activity. In the geoengineering simulation the quasi‐biennial oscillation of the tropical lower stratospheric winds remains close to the presently observed quasi‐biennial oscillation, even for large amounts of SO2 injection. Water vapor in the stratosphere increases substantially: by 25% with ∼20 Tg SO_2/year annual injection and by up to 90% with a ∼40 Tg SO_2/year injection. Stratospheric column ozone in the geoengineering simulation is predicted to recover to or supersede preozone hole conditions by the end of the century