We realize a potential platform for an efficient spin-photon interface, namely negatively-charged silicon-vacancy centers in a diamond membrane coupled to the mode of a fully-tunable, fiber-based, optical resonator. We demonstrate that introducing the thin (∼200nm), single crystal diamond membrane into the mode of the resonator does not change the cavity properties, which is one of the crucial points for an efficient spin-photon interface. In particular, we observe constantly high Finesse values of up to 3000 and a linear dispersion in the presence of the membrane. We observe cavity-coupled fluorescence from an ensemble of SiV− centers with an enhancement factor of ∼1.9. Furthermore from our investigations we extract the ensemble absorption and extrapolate an absorption cross section of (2.9±2)×10−12cm2 for a single SiV− center, much higher than previously reported.