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Since the discovery of high-transition-temperature (high-T_c) superconductivity in layered copper oxides, many researchers have searched for similar behaviour in other layered metal oxides involving 3d-transition metals, such as cobalt and nickel. Such attempts have so far failed, with the result that the copper oxide layer is thought to be essential for superconductivity. Here we report that Na_xCoO₂·yH₂O (x ≈ 0.35, y ≈ 1.3) is a superconductor with a T_c of about 5 K. This compound consists of two-dimensional CoO₂ layers separated by a thick insulating layer of Na⁺ ions and H₂O molecules. There is a marked resemblance in superconducting properties between the present material and high-T_c copper oxides, suggesting that the two systems have similar underlying physics.