We have designed and experimentally implemented a circuit of inductively-coupled superconducting charge qubits, where a Josephson junction is used as an inductance, and the coupling between the qubits is controlled by an applied magnetic flux. Spectroscopic measurements on the circuit are in good agreement with theoretical calculations. We observed anticrossings which originate from the coupling between the qubit and the plasma mode of the Josephson junction. Moreover, the size of the anticrossing depends on the external magnetic flux, which demonstrates the controllability of the coupling. Comment: Accepted for publication in PRB. 11 pages, 7 figures