Quantum computers have the capability of out-performing their classical counterparts for certain computational problems1. Several scalable quantum-computing architectures have been proposed. An attractive architecture is a large set of physically independent qubits arranged in three spatial regions where (1) the initialized qubits are stored in a register, (2) two qubits are brought together to realize a gate and (3) the readout of the qubits is carried out2, 3. For a neutral-atom-based architecture, a natural way to connect these regions is to use optical tweezers to move qubits within the system. In this letter we demonstrate the coherent transport of a qubit, encoded on an atom trapped in a submicrometre tweezer, over a distance typical of the separation between atoms in an array of optical traps4, 5, 6. Furthermore, we transfer a qubit between two tweezers, and show that this manipulation also preserves the coherence of the qubit.