  We present a divergence-free semi-implicit finite volume scheme for the
simulation of the ideal magnetohydrodynamics (MHD) equations which is stable
for large time steps controlled by the local transport speed at all Mach and
Alfv\'en numbers. An operator splitting technique allows to treat the
convective terms explicitly while the hydrodynamic pressure and the magnetic
field contributions are integrated implicitly, yielding two decoupled linear
implicit systems. The linearity of the implicit part is achieved by means of a
semi-implicit time linearization. This structure is favorable as second-order
accuracy in time can be achieved relying on the class of semi-implicit
IMplicit-EXplicit Runge-Kutta (IMEX-RK) methods. In space, implicit
cell-centered finite difference operators are designed to discretely preserve
the divergence-free property of the magnetic field on three-dimensional
Cartesian meshes. The new scheme is also particularly well suited for low Mach
number flows and for the incompressible limit of the MHD equations, since no
explicit numerical dissipation is added to the implicit contribution and the
time step is scale independent. Likewise, highly magnetized flows can benefit
from the implicit treatment of the magnetic fluxes, hence improving the
computational efficiency of the novel method. The convective terms undergo a
shock-capturing second order finite volume discretization to guarantee the
effectiveness of the proposed method even for high Mach number flows. The new
scheme is benchmarked against a series of test cases for the ideal MHD
equations addressing different acoustic and Alfv\'en Mach number regimes where
the performance and the stability of the new scheme is assessed.
