  In this paper I study the magnetosphere of a black hole that is connected by
the magnetic field to a thin conducting Keplerian disk. I consider the case of
a Schwarzschild black hole only, leaving the more interesting but difficult
case of a Kerr black hole to a future study. I assume that the magnetosphere is
ideal, stationary, axisymmetric, and force-free. I pay a special attention to
the two singular surfaces present in the system, i.e., the event horizon and
the inner light cylinder; I use the regularity condition at the light cylinder
to determine the poloidal electric current as a function of poloidal magnetic
flux. I solve numerically the Grad--Shafranov equation, which governs the
structure of the magnetosphere, for two cases: the case of a nonrotating disk
and the case of a Keplerian disk. I find that, in both cases, the poloidal flux
function on the horizon matches a simple analytical expression corresponding to
a radial magnetic field that is uniform on the horizon. Using this result, I
express the poloidal current as an explicit function of the flux and find a
perfect agreement between this analytical expression and my numerical results.
