  The exact solution for a static spherically symmetric field outside a charged
point particle is found in a non-linear $U(1)$ gauge theory with a logarithmic
Lagrangian. The electromagnetic self-mass is finite, and for a particular
relation between mass, charge, and the value of the non-linearity coupling
constant, $\lambda$, the electromagnetic contribution to the Schwarzschild mass
is equal to the total mass. If we also require that the singularity at the
origin be hidden behind a horizon, the mass is fixed to be slightly less than
the charge. This object is a {\em black point.}
