We present measurements of an optomechanical system in which the mechanical element is inside the cavity, and couples dispersively to the intracavity field. This geometry makes it easier to simultaneously achieve high optical finesse and high mechanical quality factor in an optomechanical device. We measured the linear optical properties of a such a device in which the mechanical element is a 50 nm thick silicon nitride membrane. We find that the device's finesse, resonant transmission and resonant reflection can be explained with a simple model which allows us to extract the membrane's optical loss. Our results indicate that it should be possible to increase the finesse of these devices to 5 × 105 or higher.