This is a randomized study (study drug assigned by chance) using a 2:1 randomization. It is an open-label (all people know study drug), active-controlled (comparing to a different drug used for the same condition), parallel-group (different treatment groups continue with separate treatments throughout the study), multicenter study. This study will be divided into 3 phases, screening, treatment, follow-up and optional extension phase (OEP). During screening, potential participants will be assessed for study eligibility after providing signed informed consent. Approximately 570 patients who satisfy all inclusion and exclusion criteria will be randomly assigned in a 2:1 ratio to either the trabectedin (n=380) or dacarbazine (n=190) treatment groups. During the treatment phase, patients will receive study drug once every 3 weeks, until disease progression (defined by Response Evaluation Criteria in Solid Tumors [RECIST] Version 1.1 criteria) or signs of toxicity. Assessments will be performed to evaluate the effectiveness of the drug, and patient safety will be monitored. During the follow-up phase, after the last dose of study drug, clinical outcomes for patients will be evaluated. Trabectedin will be administered at a dose of 1.5 milligram per square meters (mg/m^2) through a catheter into a large vein as a 24-hour intravenous (IV) infusion, once every 3 weeks, until disease progression or signs of toxicity. Dacarbazine will be administered at a dose of 1.0 g/m^2 as a 20-120 minute infusion, once every 3 weeks, until disease progression or signs of toxicity. In the OEP, participants who were previously randomized to the dacarbazine group will have the option to receive trabectedin at the discretion of the investigator.