Kerr-Newman and genuine Kerr black holes in the Bertotti-Robinson magnetic field

We present a spacetime that describes a rotating and charged black hole immersed in an external magnetic field. This exact solution to the Einstein-Maxwell equations has a surprisingly simple form which reduces to standard Kerr-Newman metric, and to the uniform Bertotti-Robinson magnetic universe. We show that the Kerr-BR solution presented in [Phys. Rev. Lett. {\bf 135}, 181401 (2025)] has a special charge, and we identify a genuine Kerr black hole without any charge, in the magnetic field. In the weak-field limit it gives the Wald solution. We analyze main physical properties of the new family, namely singularities, horizons, regularity of axes, charges, entropy and temperature.