The monolithic integration of novel nanomaterials with mature and established technologies has considerably widened the scope and potential of nanophotonics. For example, the integration of single semiconductor quantum dots into photonic crystals has enabled highly efficient single-photon sources. Recently, there has also been an increasing interest in using graphene - a single atomic layer of carbon - for optoelectronic devices. However, being an inherently weak optical absorber (only 2.3 % absorption), graphene has to be incorporated into a high-performance optical resonator or waveguide to increase the absorption and take full advantage of its unique optical properties. Here, we demonstrate that by monolithically integrating graphene with a Fabry-Perot microcavity, the optical absorption is 26-fold enhanced, reaching values >60 %. We present a graphene-based microcavity photodetector with record responsivity of 21 mA/W. Our approach can be applied to a variety of other graphene devices, such as electro-absorption modulators, variable optical attenuators, or light emitters, and provides a new route to graphene photonics with the potential for applications in communications, security, sensing and spectroscopy.