We have used resonant x-ray scattering to determine the momentum dependent charge correlations in YBa2Cu3O6.55 samples with highly ordered chain arrays of oxygen acceptors (ortho-II structure). The results reveal nearly critical, biaxial charge density wave (CDW) correlations at in-plane wave vectors (0.315, 0) and (0, 0.325). The corresponding scattering intensity exhibits a strong uniaxial anisotropy. The CDW amplitude and correlation length are enhanced as superconductivity is weakened by an external magnetic field. Analogous experiments were carried out on a YBa2Cu3O6.6 crystal with a dilute concentration of spinless (Zn) impurities, which had earlier been shown to nucleate incommensurate magnetic order. Compared to pristine crystals with the same doping level, the CDW amplitude and correlation length were found to be strongly reduced. These results indicate a three-phase competition between spin-modulated, charge-modulated, and superconducting states in underdoped YBa2Cu3O 6+δ .PACS numbers: 74.20. Rp, 74.25.Gz, 74.25.Kc, 74.72.Bk High-temperature superconductivity in the cuprates arises from doping of charge carriers into Mott-insulators with antiferromagnetically ordered CuO 2 planes.[1] At sufficiently high density, the carriers screen out the random potential created by the donor or acceptor atoms and generate a uniform metallic state out of which superconductivity arises. In underdoped cuprates, however, the screening is less effective, and the role of materialsspecific disorder in the formation of the unusual spin and/or charge textures observed in this regime of the phase diagram has been a subject of long-standing debate. [2-16] Recent research on the stoichiometric underdoped compounds YBa 2 Cu 3 O 6.5 and YBa 2 Cu 4 O 8 has provided new perspectives for the resolution of the influence of disorder on the electronic phase behavior of the underdoped cuprates. In these materials, the oxygen acceptors are arranged in ordered chains rather than placed randomly in the crystal lattice, so that chemical and structural disorder is minimized.[17] The results of recent quantum oscillation experiments [18][19][20] in high magnetic fields indicate a reconstruction of the Fermi surface by a long-range electronic superstructure. [21][22][23] This discovery has sparked another intense debate on the nature of the high-field ordering and its relation to the "pseudogap" observed in these and other underdoped cuprates [24] above the superconducting transition temperature, T c , in the absence of external fields. The pseudogap, in turn, is intimately related to the superconducting gap, and an explanation of its origin is considered an essential element of any theory of high-temperature superconductivity.Whereas research on YBa 2 Cu 4 O 8 has been limited, because only small crystals are available and the doping level cannot be varied in a straightforward manner, YBa 2 Cu 3 O 6.5 is a member of the extensively studied YBa 2 Cu 3 O 6+δ (YBCO 6+δ , 123) family, where the concentration of mobile holes in the CuO 2 layer...