There is increasing evidence that inhomogeneous distributions of charge and spin-so-called ＇striped phases＇-play an important role in determining the properties of the high-temperature superconductors. For example, recent neutron-scattering measurements on the YBa2Cu3O7-x family of materials show both spin and charge fluctuations that are consistent with the striped-phase picture. But the fluctuations associated with a striped phase are expected to be one-dimensional, whereas the magnetic fluctuations observed to date appear to display two-dimensional symmetry. We show here that this apparent two-dimensionality results from measurements on twinned crystals, and that similar measurements on substantially detwinned crystals of YBa2Cu3O6.6 reveal the one-dimensional character of the magnetic fluctuations, thus greatly strengthening the striped-phase interpretation. Moreover, our results also suggest that superconductivity originates in charge stripes that extend along the b crystal axis, where the superfluid density is found to be substantially larger than for the a direction.