  Cavity optomechanics, where photons are coupled to mechanical motion,
provides the tools to control mechanical motion near the fundamental quantum
limits. Reaching single-photon strong coupling would allow to prepare the
mechanical resonator in non-Gaussian quantum states. Preparing massive
mechanical resonators in such states is of particular interest for testing the
boundaries of quantum mechanics. This goal remains however challenging due to
the small optomechanical couplings usually achieved with massive devices. Here
we demonstrate a novel approach where a mechanical resonator is magnetically
coupled to a microwave cavity. We measure a single-photon coupling of $g_0/2
\pi \sim 3$ kHz, an improvement of one order of magnitude over current
microwave optomechanical systems. At this coupling we measure a large
single-photon cooperativity with $C_0 \gtrsim 10$, an important step to reach
single-photon strong coupling. Such a strong interaction allows us to cool the
massive mechanical resonator to a third of its steady state phonon population
with less than two photons in the microwave cavity. Beyond tests for quantum
foundations, our approach is also well suited as a quantum sensor or a
microwave to optical transducer.
