Demonstrating and exploiting the quantum nature of macroscopic mechanical objects would help us to investigate directly the limitations of quantum-based measurements and quantum information protocols, as well as to test long-standing questions about macroscopic quantum coherence(1-3). Central to this effort is the necessity of long-lived mechanical states. Previous efforts have witnessed quantum behaviour(4), but for a low-quality-factor mechanical system. The field of cavity optomechanics and electromechanics(5,6), in which a high-quality-factor mechanical oscillator is parametrically coupled to an electromagnetic cavity resonance, provides a practical architecture for cooling, manipulation and detection of motion at the quantum level(1). One requirement is strong coupling(7-9), in which the interaction between the two systems is faster than the dissipation of energy from either system. Here, by incorporating a free-standing, flexible aluminium membrane into a lumped-element superconducting resonant cavity, we have increased the single-photon coupling strength between these two systems by more than two orders of magnitude, compared to previously obtained coupling strengths. A parametric drive tone at the difference frequency between the mechanical oscillator and the cavity resonance dramatically increases the overall coupling strength, allowing us to completely enter the quantum-enabled, strong-coupling regime. This is evidenced by a maximum normal-mode splitting of nearly six bare cavity linewidths. Spectroscopic measurements of these 'dressed states' are in excellent quantitative agreement with recent theoretical predictions(10,11). The basic circuit architecture presented here provides a feasible path to ground-state cooling and subsequent coherent control and measurement of long-lived quantum states of mechanical motion.