  We experimentally demonstrate coupling of an atomically thin, free-standing
graphene membrane to an optical cavity. By changing the position of the
membrane along the standing-wave field of the cavity we tailor the dissipative
coupling between the membrane and the cavity, and we show that the dissipative
coupling can outweigh the dispersive coupling. Such a system, for which
controlled dissipation prevails dispersion, will prove useful for novel
laser-cooling schemes in optomechanics. In addition, we have determined the
continuous-wave optical damage threshold of free-standing monolayer graphene of
1.8(4)~MW/cm$^2$ at 780nm.
