We present experimental and theoretical results on changing the fluorescence emission spectrum of a single molecule by embedding it within a tunable planar microcavity with subwavelength spacing. The cavity length is changed with nanometer precision by using a piezoelectric actuator. By varying its length, the local mode structure of the electromagnetic field is changed together with the radiative coupling of the emitting molecule to the field. Because mode structure and coupling are both frequency dependent, this leads to a renormalization of the emission spectrum of the molecule. We develop a theoretical model for these spectral changes and find excellent agreement between theoretical prediction and experimental results.