The acceleration of the Brewer-Dobson circulation is one of the most robust impacts on the atmospheric circulation of increasing levels of carbon dioxide (CO2). However, a complete understanding of the mechanisms leading to that acceleration is as yet incomplete. Here, using a single-model framework, we separate and quantify three largely independent pathways that lead to BDC acceleration under an abrupt 4xCO2 forcing: the warming of sea surface temperatures (SSTs), the cooling of the stratosphere from direct radiative forcing, and the composition feedbacks associated with changes of the ozone layer, each of which is caused by increased CO2. We accomplish this by contrasting NASA GISS Model E2.2 simulations in fully-coupled and atmosphere-only configurations. First, we validate our methodology, and demonstrate the response in the fully-coupled model can be simulated as the linear sum of contributions from warmer SSTs, direct radiative effects, and ozone changes. Second, we show that while surface warming induces ~85% of the BDC acceleration, its impact is limited to the lower stratosphere. By comparison, in the upper-and-middle stratosphere, the BDC response is dominated by changes due to direct radiative forcing from CO2 (80% of the acceleration at 10 hPa). Third, we find that changes in ozone lead to a deceleration of the BDC, nearly canceling the acceleration by the CO2 direct radiative forcing in the mid-to-lower stratosphere (30-70 hPa).