The pseudogap region of the phase diagram is an important unsolved puzzle in the field of high-transition-temperature (high-Tc) superconductivity, characterized by anomalous physical properties1, 2. There are open questions about the number of distinct phases and the possible presence of a quantum-critical point underneath the superconducting dome3, 4, 5. The picture has remained unclear because there has not been conclusive evidence for a new type of order. Neutron scattering measurements for YBa2Cu3O6+ (YBCO) resulted in contradictory claims of no6, 7 and weak8, 9 magnetic order, and the interpretation of muon spin relaxation measurements on YBCO10, 11 and of circularly polarized photoemission experiments on Bi2Sr2CaCu2O8+(refs 12, 13) has been controversial. Here we use polarized neutron diffraction to demonstrate for the model superconductor HgBa2CuO4+ (Hg1201) that the characteristic temperature T* marks the onset of an unusual magnetic order. Together with recent results for YBCO14, 15, this observation constitutes a demonstration of the universal existence of such a state. The findings appear to rule out theories that regard T* as a crossover temperature16, 17, 18 rather than a phase transition temperature19, 20, 21. Instead, they are consistent with a variant of previously proposed charge-current-loop order19, 20 that involves apical oxygen orbitals22, and with the notion that many of the unusual properties arise from the presence of a quantum-critical point3, 4, 5, 19.