We present a combined neutron diffraction and bulk thermodynamic study of the natural mineral linarite PbCuSO4(OH)(2), this way establishing the nature of the ground-state magnetic order. An incommensurate magnetic ordering with a propagation vector k = (0, 0.186, 1/2) was found below T-N = 2.8 K in a zero magnetic field. The analysis of the neutron diffraction data yields an elliptical helical structure, where one component (0.638 mu(B)) is in the monoclinic ac plane forming an angle with the a axis of 27(2)degrees, while the other component (0.833 mu(B)) points along the b axis. From a detailed thermodynamic study of bulk linarite in magnetic fields up to 12 T, applied along the chain direction, a very rich magnetic phase diagram is established, with multiple field-induced phases, and possibly short-range-order effects occurring in high fields. Our data establish linarite as a model compound of the frustrated one-dimensional spin chain, with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions. Long-range magnetic order is brought about by interchain coupling 1 order of magnitude smaller than the intrachain coupling. © 2012, American Physical Society.