We compute the current-voltage (I-V) characteristic of a mesoscopic Josephson junction, formed from two weakly linked, superconducting dots. A key feature of such structures is the presence of quasi-particle resonances in the current-phase relation. We show that these lead to measurable changes in the I-V characteristic compared with conventional macroscopic junctions. For an overdamped junction, the I-V characteristic is predicted to shift to higher values of the current, but the critical current is unchanged. For an underdamped junction exhibiting hysteresis, this shift is accompanied by an increase in the critical current associated with the lower branch of the I-V curve.