Materials that can in principle exhibit both superconductivity and ferromagnetism are caught in a dilemma : both states represent long-range order, but are in general mutually exclusive. When the material favours a ground state with a large magnetic moment, as is the case for Er4Rh4B (ref. 1), superconductivity is destroyed. For superconductivity to persist, the magnetic structure would need to adopt an antiferromagnetic modulation of short enough wavelength to ensure a small net moment on the length scale of the superconducting coherence length. The intermetallic-borocarbide superconductors(2-4) RNi2B2C (where R is a rare-earth element) have shed new light on this balance between magnetism and superconductivity. The response of these materials in the superconducting state to a magnetic field is dominated by the formation of a flux-line lattice-a regular array of quantized magnetic vortices whose symmetry and degree of order are easily modified and thus can be expected to interact with an underlying magnetic modulation. In TmNi2B2C, superconductivity and antiferromagnetic modulated ordering coexist below 1.5 K (refs 5-7). Here we present the results of a small-angle neutron-scattering study of this compound which show that the structure of the magnetic modulation and the symmetry of the flux-line lattice are intimately coupled, resulting in a complex phase diagram.