The double-gamma (gamma gamma)-decay of a quantum system in an excited state is a fundamental second-order process of quantum electrodynamics. In contrast to the well-known single-gamma (gamma)-decay, the gamma-decay is characterized by the simultaneous emission of two gamma quanta, each with a continuous energy spectrum. In nuclear physics, this exotic decay mode has only been observed for transitions between states with spin-parity quantum numbers J(pi) = 0(+) (refs 1-3). Single-gamma decays the main experimental obstacle to observing the yy-decay are strictly forbidden for these 0(+) -> 0(+) transitions. Here we report the observation of the gamma-decay of an excited nuclear state (J(pi) = 11/2(-)) that is directly competing with an allowed y-decay (to ground state J(pi) = 3/2(+)). The branching ratio of the competitive gamma gamma-decay of the 11/2- isomer of Ba-137 to the ground state relative to its single gamma-decay was determined to be (2.05 +/- 0.37) X 10(-6). From the measured angular correlation and the shape of the energy spectra of the individual gamma-rays, the contributing combinations of multipolarities of the gamma radiation were determined. Transition matrix elements calculated using the quasipartide-phonon model reproduce our measurements well. The gamma gamma-decay rate gives access to so far unexplored important nuclear structure information, such as the generalized (off-diagonal) nuclear electric polarizabilities and magnetic susceptibilities(3).