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In this paper, we will provide a feasible mission design for a multiple-rendezvous mission to Jupiter's Trojans. It is based on solar electric propulsion, as being currently used on the DAWN spacecraft, and other flight-proven technology. First, we have selected a set of mission objectives, the prime objective being the detection of water -especially subsurface water -to provide evidence for the Trojans' formation at large solar distances. Based on DAWN and other comparable missions, we have determined suitable payload instruments to achieve these objectives. Afterwards, we have designed a spacecraft that is able to carry the selected payload to the Trojan region and rendezvous successively with three target bodies within a maximum mission duration of 15 years. Accurate low-thrust trajectories have been obtained with a global low-thrust trajectory optimization program (InTrance). During the transfer from Earth to the first target, the spacecraft is propelled by two RIT-22 ion engines from EADS Astrium, whereas a single RIT-15 is used for transfers within the Trojan region to reduce the required power. For power generation, the spacecraft uses a multi-junction solar array that is supported by concentrators. To achieve moderate mission costs, we have restricted the launch mass to a maximum of 1600 kg, the maximum interplanetary injection capability of a Soyuz/Fregat launcher. Our final layout has a mass of 1400 kg, yielding a margin of about 14%. Nestor (a member of the L4-population) was determined as the first mission target. It can be reached within 4.6 years from launch. The fuel mass ratio for this transfer is about 35%. The stay time at Nestor is 1.2 years. Eurymedon was selected as the second target (transfer time 3.5 years, stay time 3.0 years) and Irus as the third target (transfer time 2.2 years). The transfers within the Trojan L4-population can be accomplished with fuel mass ratios of about 3% for each trajectory leg. Including the stay times in orbit around the targets, the mission can be accomplished within a total duration of about 14.5 years. According to our mission analysis, it is also feasible to fly to the L5-population with similar flight times. It has to be noted that -for a first analysis -we have taken only the named targets into account. Allowing also rendezvous with unnamed objects will very likely decrease the mission duration. Based on a scaling of DAWN's mission costs (due to comparable scientific instruments and mission objectives), and taking into account the longer mission duration and the potential re-use of already developed technology, we have estimated that these three rendezvous can be accomplished with a budget of about 250 Million Euros, i.e. about 25% of ROSETTA's budget.