  We demonstrate large normal-mode splitting between a magnetostatic mode (the
Kittel mode) in a ferromagnetic sphere of yttrium iron garnet and a microwave
cavity mode. Strong coupling is achieved in the quantum regime where the
average number of thermally or externally excited magnons and photons is less
than one. We also confirm that the coupling strength is proportional to the
square root of the number of spins. A nonmonotonic temperature dependence of
the Kittel-mode linewidth is observed below 1 K and is attributed to the
dissipation due to the coupling with a bath of two-level systems.
