Inclusion Criteria:

- Histologically proven advanced-relapsed epithelial ovarian, primary peritoneal, or
fallopian tube cancer

- Eastern Cooperative Oncology Group (ECOG) performance status grade of 0 or 1

- Received first-line treatment with a platinum-based regimen and had no evidence of
disease progression for 6 months after the last dose

- Received second-line treatment with a platinum-based regimen, with progression of
disease after attaining a complete response (CR) or partial response (PR)

- Progression of disease based on imaging after the second-line platinum-based regimen
(individuals treated with a pegylated liposomal doxorubicin-containing regimen as a
second-line therapy are eligible if subsequent disease progression occurs >=9 months
from the first dose)

- Evidence of measurable disease at screening as evaluated by Response Evaluation
Criteria in Solid Tumors (RECIST) (Version 1.1)

- Able to receive intravenous (IV) dexamethasone or an equivalent IV corticosteroid

- Have a known BRCA 1/2 mutation status (for participants who do not have a known BRCA
1/2 status at screening, a blood sample will be collected to determine the status
with the results available prior to randomization

- Laboratory values within protocol -defined parameters

- Have left ventricular ejection fraction by multigated acquisition scan (MUGA) scan or
2D-ECHO within normal limits for the institution

- Have side effects (except alopecia) of prior treatment resolved to at least Grade 1
according to the National Cancer Institute - Common Terminology Criteria of Adverse
Events (NCICTCAE) (Version 4.0)

- Have a negative urine or serum pregnancy test at screening

- Agrees to protocol-defined use of effective contraception

Exclusion Criteria:

- Diagnosis of ovarian carcinoma with mucinous histology

- Had more than 2 prior lines of chemotherapy

- Prior exposure to trabectedin or hypersensitivity to any of the excipients

- Prior treatment with doxorubicin or other anthracycline at cumulative doses greater
than 300 mg/m2 (calculated using doxorubicin equivalent doses: 1 mg doxorubicin = 1
mg Doxil/Caelyx = 1.8 mg epirubicin = 0.3 mg mitoxantrone = 0.25 mg idarubicin)

- Unwilling or unable to have a central venous catheter placed

- Pregnant or breast-feeding

- Less than 3 weeks from radiation therapy, experimental therapy, hormonal therapy,
prior chemotherapy, or biological therapy

- History of another neoplastic disease (except non-metastatic basal cell carcinoma or
squamous cell carcinoma of the skin or cervical carcinoma in situ adequately treated)
unless in remission for 5 years

- Known allergies, hypersensitivity, or intolerance to Doxil, dexamethasone, or their
excipients

- Known history of central nervous system metastasis

- Known significant chronic liver disease, such as cirrhosis or active hepatitis
(potential participants who test positive for hepatitis B surface antigen or
hepatitis C antibodies are allowed provided they do not have active disease requiring
antiviral therapy)

- Had a myocardial infarct within 6 months before enrollment, New York Heart
Association (NYHA) Class II or greater heart failure, uncontrolled angina, severe
uncontrolled ventricular arrhythmias, clinically significant pericardial disease, or
electrocardiographic evidence of acute ischemic or active conduction system
abnormalities

- Has any of the following medical conditions: uncontrolled diabetes, psychiatric
disorder (including dementia) that prevents compliance with protocol, uncontrolled
seizures, newly diagnosed deep vein thrombosis, active systemic infection that is
likely to interfere with study procedure or results

- Has any condition that, in the opinion of the investigator, would compromise the
well-being of the participant or the study or prevent the participant from meeting or
performing study requirements