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Muon spin rotation measurements in ultra-clean single crystals of heavily underdoped superconducting YBa2Cu3O6+x (x approximate to 0.365) are presented. The material shows a sharp superconducting transition below T = 15 K. By field cooling and shifting the applied field, we show that the superconducting state pins magnetic flux and develops a flux lattice below T-c, indicating that the superconducting state exists throughout the sample on a microscopic scale. At the same temperature, a disordered magnetic state appears on a nanoscale with at least one well-defined internal field probed by the muon. These two states coexist on a nanometer lengthscale and over a narrow region of oxygen doping. (c) 2005 Elsevier B.V. All rights reserved.