Entanglement is one of the most fundamental properties of quantum mechanics(1-3), and is the key resource for quantum information processing(4,5) (QIP). Bipartite entangled states of identical particles have been generated and studied in several experiments, and post-selected or heralded entangled states involving pairs of photons, single photons and single atoms, or different nuclei in the solid state, have also been produced(6-12). Here we use a deterministic quantum logic gate to generate a 'hybrid' entangled state of two trapped-ion qubits held in different isotopes of calcium, perform full tomography of the state produced, and make a test of Bell's inequality with non-identical atoms. We use a laser-driven two-qubit gate(13), whose mechanism is insensitive to the qubits' energy splittings, to produce a maximally entangled state of one Ca-40(+) qubit and one Ca-43(+) qubit, held 3.5 micrometres apart in the same ion trap, with 99.8 +/- 0.6 per cent fidelity. We test the CHSH (Clauser-Horne-Shimony-Holt)(14) version of Bell's inequality for this novel entangled state and find that it is violated by 15 standard deviations; in this test, we close the detection loophole(8) but not the locality loophole(7). Mixed-species quantum logic is a powerful technique for the construction of a quantum computer based on trapped ions, as it allows protection of memory qubits while other qubits undergo logic operations or are used as photonic interfaces to other processing units(15,16). The entangling gate mechanism used here can also be applied to qubits stored in different atomic elements; this would allow both memory and logic gate errors caused by photon scattering to be reduced below the levels required for fault-tolerant quantum error correction, which is an essential prerequisite for general-purpose quantum computing.