Superconductivity and magnetism generally do not coexist. Changing the relative number of up and down spin electrons disrupts the basic mechanism of superconductivity, where atoms of opposite momentum and spin form Cooper pairs. Nearly forty years ago Fulde and Ferrell and Larkin and Ovchinnikov (FFLO) proposed an exotic pairing mechanism in which magnetism is accommodated by the formation of pairs with finite momentum. Despite intense theoretical and experimental efforts, however, polarized superconductivity remains largely elusive. Unlike the three-dimensional (3D) case, theories predict that in one dimension (1D) a state with FFLO correlations occupies a major part of the phase diagram. Here we report experimental measurements of density profiles of a two-spin mixture of ultracold 6 Li atoms trapped in an array of 1D tubes (a system analogous to electrons in 1D wires). At finite spin imbalance, the system phase separates with an inverted phase profile, as compared to the 3D case. In 1D, we find a partially polarized core surrounded by wings which, depending on the degree of polarization, are composed of either a completely paired or a fully polarized Fermi gas. Our work paves the way to direct observation and characterization of FFLO pairing. © 2010 Macmillan Publishers Limited. All rights reserved.