  We consider a stationary axisymmetric force-free degenerate magnetosphere of
a rotating Kerr black hole surrounded by a thin Keplerian infinitely-conducting
accretion disk. We focus on the closed-field geometry characterize by a direct
magnetic coupling between the disk and the hole's event horizon. We first argue
that the hole's rotation necessarily limits the radial extent of the force-free
link on the disk surface: the faster the hole rotates, the smaller the
magnetically-connected inner part of the disk has to be. We then show that this
is indeed the case by solving numerically the Grad--Shafranov equation--the
main differential equation describing the structure of the magnetosphere. An
important element in our approach is the use of the regularity condition at the
inner light cylinder to fix the poloidal current as a function of the poloidal
magnetic flux. As an outcome of our computations, we are able to chart out the
maximum allowable size of the portion of the disk that is magnetically
connected to the hole as a function of the black hole spin. We also calculate
the angular momentum and energy transfer between the hole and the disk that
takes place via the direct magnetic link. We find that both of these quantities
grow rapidly and that their deposition becomes highly concentrated near the
inner edge of the disk as the black hole spin is increased.
