Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. Although discrete time crystals have been extensively studied in periodically driven systems, the experimental realization of a continuous time crystal is still pending. We report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system that is characterized by emergent oscillations in the intracavity photon number. The phase of the oscillation was found to be random for different realizations, and hence, this dynamical many-body state breaks continuous time translation symmetry spontaneously. Furthermore, the observed limit cycles are robust against temporal perturbations and therefore demonstrate the realization of a continuous time crystal.