We propose a quantum computer architecture that is robust against decoherence and scalable. As a qubit we adopt rotational states of a nonpolar ionic molecule trapped in an ion trap. It is revealed that the rotational-state qubits are much more immune to decoherence than the conventional electronic-state qubits of atomic ions. A complete method set that includes state preparation, a single-qubit gate, a controlled-NOT gate, and qubit readout suitable for the rotational-state qubits is provided. Since the ionic molecules can be transported in an array of ion traps, the rotational-state qubits are expected to be a promising candidate to build a large-scale quantum computer. DOI: 10.1103/PhysRevA.87.040302