The double-gamma (γγ)-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 (γ)-decay, the γγ-decay is characterized by the simultaneous emission of two γ 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(π) = 0(+) (refs 1-3). Single-gamma decays-the main experimental obstacle to observing the γγ-decay-are strictly forbidden for these 0(+) → 0(+) transitions. Here we report the observation of the γγ-decay of an excited nuclear state (J(π) = 11/2(-)) that is directly competing with an allowed γ-decay (to ground state J(π) = 3/2(+)). The branching ratio of the competitive γγ-decay of the 11/2(-) isomer of (137)Ba to the ground state relative to its single γ-decay was determined to be (2.05 ± 0.37) × 10(-6). From the measured angular correlation and the shape of the energy spectra of the individual γ-rays, the contributing combinations of multipolarities of the γ radiation were determined. Transition matrix elements calculated using the quasiparticle-phonon model reproduce our measurements well. The γγ-decay rate gives access to so far unexplored important nuclear structure information, such as the generalized (off-diagonal) nuclear electric polarizabilities and magnetic susceptibilities.