The alternating- and direct-current (a.c. and d.c.) Josephson effects were first discovered in a system of two superconductors, the macroscopic wavefunctions of which are weakly coupled via a tunnelling barrier. In the a.c. Josephson effect, a constant chemical potential difference (voltage) is applied, which causes an oscillating current to flow through the barrier. Because the frequency is proportional to the chemical potential difference only, the a.c. Josephson effect serves as a voltage standard. In the d.c. Josephson effect, a small constant current is applied, resulting in a constant supercurrent flowing through the barrier. In a sense, the particles do not 'feel' the presence of the tall tunnelling barrier, and flow freely through it with no driving potential. Bose-Einstein condensates should also support Josephson effects; however, while plasma oscillations have been seen in a single Bose-Einstein condensate Josephson junction, the a.c. Josephson effect remains elusive. Here we observe the a.c. and d.c. Josephson effects in a single Bose-Einstein condensate Josephson junction. The d.c. Josephson effect has been observed previously only in superconducting systems; in our study, it is evident when we measure the chemical potential-current relation of the Bose-Einstein condensate Josephson junction. Our system constitutes a trapped-atom interferometer with continuous readout, which operates on the basis of the a.c. Josephson effect. In addition, the measured chemical potential-current relation shows that the device is suitable for use as an analogue of the superconducting quantum interference device, which would sense rotation.