  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
super-conductivity. Neutron scattering measurements of spin fluctuations in
hole-doped copper oxides have revealed an unusual `hour-glass' feature in the
momentum-resolved magnetic spectrum, 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, an idea supported by measurements on stripe-ordered
La1.875Ba0.125CuO4. However, many copper oxides without stripe order 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 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.
