  We revisit the Blandford & Znajek (1977) process and solve the fundamental
equation that governs the structure of the steady-state force-free
magnetosphere around a Kerr black hole. The solution depends on the
distributions of the magnetic field angular velocity omega and the poloidal
electric current I. These are not arbitrary. They are determined
self-consistently by requiring that magnetic field lines cross smoothly the two
singular surfaces of the problem, the inner `light surface' located inside the
ergosphere, and the outer `light surface' which is the generalization of the
pulsar light cylinder. We find the solution for the simplest possible magnetic
field configuration, the split monopole, through a numerical iterative
relaxation method analogous to the one that yields the structure of the
steady-state axisymmetric force-free pulsar magnetosphere (Contopoulos, Kazanas
& Fendt 1999). We obtain the rate of electromagnetic extraction of energy and
confirm the results of Blandford and Znajek and of previous time dependent
simulations. Furthermore, we discuss the physical applicability of magnetic
field configurations that do not cross both `light surfaces'.
