  We derive from first principles the mechanical pressure $P$, defined as the
force per unit area on a bounding wall, in a system of spherical, overdamped,
active Brownian particles at density $\rho$. Our exact result relates $P$, in
closed form, to bulk correlators and shows that (i) $P(\rho)$ is a state
function, independent of the particle-wall interaction; (ii) interactions
contribute two terms to $P$, one encoding the slow-down that drives
motility-induced phase separation, and the other a direct contribution well
known for passive systems; (iii) $P(\rho)$ is equal in coexisting phases. We
discuss the consequences of these results for the motility-induced phase
separation of active Brownian particles, and show that the densities at
coexistence do not satisfy a Maxwell construction on $P$.
