Neutron scattering is used to characterize the magnetism of the vortices for the optimally doped high-temperature superconductor La2-xSrxCuO4 (x = 0.163) in an applied magnetic field. As temperature is reduced, Low-frequency spin fluctuations first disappear with the loss of vortex mobility, but then reappear. We find that the vortex state can be regarded as an inhomogeneous mixture of a superconducting spin fluid and a material containing a nearly ordered antiferromagnet. These experiments show that as for many other properties of cuprate superconductors, the important underlying microscopic forces are magnetic.