For the geometry of a spherical droplet carrying ionic charges, a solution to the Poisson-Boltzmann equation can be found via two routes: by numerical simulation and as an algebraic series. The maximum charge is limited by the surface tension of the liquid according to Rayleigh's law. Potential and ionic concentration profiles have been determined for a variety of electrolytes in aqueous solution, emphasizing charges of a magnitude that just meet the Rayleigh criterion for several droplet radii. The charge is found to occupy a diffuse layer, contrasting with the frequent assumption that the charge resides on the droplet's surface. Under conditions that have been delineated, the charge consists solely of ions sharing the sign of the droplet's charge, counterions being absent.