A quantum magnet, LiCuSbO4, with chains of edge-sharing spin-1/2 CuO6 octahedra is reported. While short-range order is observed for T<10 K, no zero-field phase transition or spin freezing occurs down to 100 mK. Specific heat indicates a distinct high-field phase near the 12 T saturation field. Neutron scattering shows incommensurate spin correlations with q=(0.47±0.01)π/a and places an upper limit of 70 μeV on any spin gap. Exact diagonalization of 16-spin easy-plane spin-1/2 chains with competing ferro- and antiferromagnetic interactions (J1=-75 K, J2=34 K) accounts for the T>2 K data.