This study, will take a subject's "T cells" and "teach" them to be able to recognize and attack the ovarian cancer cells. This is done by putting in a gene or genetic material that will change how a subject's T cells work and hopefully get them to attack and kill ovarian cancer cells. These new T cells are called "engineered T cells" because the new gene is causing them to become directed toward the ovarian cancer cells rather than their usual targets. These are also called "gene-modified T cells". For subjects who have the HLA A2 tissue-type marker, the T cells would be engineered to recognize a substance called "NY-ESO-1". After putting this new gene in T cells (a procedure called "gene therapy") the investigators will grow the cells in the laboratory and give these cells back to subjects.