  We have measured the spatial distribution of motile Escherichia coli inside
spherical water droplets emulsified in oil. At low cell concentrations, the
cell density peaks at the water-oil interface; at increasing concentration, the
bulk of each droplet fills up uniformly while the surface peak remains.
Simulations and theory show that the bulk density results from a `traffic' of
cells leaving the surface layer, increasingly due to cell-cell scattering as
the surface coverage rises above $\sim 10\%$. Our findings show similarities
with the physics of a rarefied gas in a spherical cavity with attractive walls.
