Quantum teleportation1provides a means to transport quantum information efficiently from one location to another, without the physical transfer of the associated quantum-information carrier. This is achieved by using the non-local correlations of previously distributed, entangled quantum bits (qubits). Teleportation is expected to play an integral role in quantum communication2and quantum computation3. Previous experimental demonstrations have been implemented with optical systems that used both discrete and continuous variables4-9, and with liquid-state nuclear magnetic resonance10. Here we report unconditional teleportation5of massive particle qubits using atomic (9Be+) ions confined in a segmented ion trap, which aids individual qubit addressing. We achieve an average fidelity of 78 per cent, which exceeds the fidelity of any protocol that does not use entanglement11. This demonstration is also important because it incorporates most of the techniques necessary for scalable quantum information processing in an ion-trap system12,13.