Cavity quantum electrodynamics (QED) systems allow the study of a variety of fundamental quantum-optics phenomena, such as entanglement, quantum decoherence and the quantum-classical boundary. Such systems also provide test beds for quantum information science. Nearly all strongly coupled cavity QED experiments have used a single atom in a high-quality-factor (high-Q) cavity. Here we report the experimental realization of a strongly coupled system in the solid state: a single quantum dot embedded in the spacer of a nanocavity, showing vacuum-field Rabi splitting exceeding the decoherence linewidths of both the nanocavity and the quantum dot. This requires a small-volume cavity and an atomic-like two-level system. The photonic crystal slab nanocavity--which traps photons when a defect is introduced inside the two-dimensional photonic bandgap by leaving out one or more holes--has both high Q and small modal volume V, as required for strong light-matter interactions. The quantum dot has two discrete energy levels with a transition dipole moment much larger than that of an atom, and it is fixed in the nanocavity during growth.