Superconductivity in layered copper oxide compounds emerges when charge carriers are added to antiferromagnetically ordered CuO₂ layers. The carriers destroy the antiferromagnetic order, but strong spin fluctuations persist throughout the superconducting phase and are intimately linked to superconductivity. Neutron scattering measurements of spin fluctuations in hole-doped copper oxides have revealed an unusual 'hour-glass' feature in the momentum-resolved magnetic spectrum that is present in a wide range of superconducting and non-superconducting materials. There is no widely accepted explanation for this feature. One possibility is that it derives from a pattern of alternating spin and charge stripes, and this idea is supported by measurements on stripe-ordered La_1.875 Ba_0.125 CuO₄ (ref. 15). Many copper oxides without stripe order, however, also exhibit an hour-glass spectrum. Here we report the observation of an hour-glass magnetic spectrum in a hole-doped antiferromagnet from outside the family of superconducting copper oxides. Our system has stripe correlations and is an insulator, which means that its magnetic dynamics can conclusively be ascribed to stripes. The results provide compelling evidence that the hour-glass spectrum in the copper oxide superconductors arises from fluctuating stripes.